GIFT  OF 
Mrs.   F.J.B.   Cordeiro 


r 


PUBLICATIONS  BY  THE  AUTHOR 

SPON   &   CHAMBERLAIN,   NEW 

YORK 

[EIGHTS. 

BAROMEIRICAL  DETERMINATION  OF   B 

30.50 

THE  ATMOSPHERE. 

A  Manual  of  Meteorology 

32.50 

THE   GYROSCOPE. 

Theory  and  Applications 

31.50 

MECHANICS  OF  ELECTRICITY. 

What  is  Electricity 

31.25 

PRINCIPLES  OF  NATURAL  PHILOSOPHY. 

32.50 

PRINCIPLES  OF 
NATURAL  PHILOSOPHY 


BY 

F.  J.  B.  CORDEIRO 

AUTHOR  OF 
'MECHANICS     OF     ELECTRICITY,"     "THE     GYROSCOPE," 
'THE  ATMOSPHERE,"  "BAROMETRICAL  HEIGHTS,"   ETC. 


NEW  YORK: 
SPON  &  CHAMBERLAIN,   123  Liberty  Street 

LONDON: 
E.  &  F.  N.  SPON,  Limited,  57  Haymarket,  S.W. 

1917 


--<? 


\r 


Pinkham  Press 
Boston 


PREFACE 

Natural  philosophy  is  the  study  of  nature,  or  the  uni- 
verse. Experience  shows  that  all  effects  are  preceded  by 
causes,  and  under  identical  circumstances  all  changes  are 
always  effected  in  the  same  manner.  Nature  is  in  a  con- 
tinual state  of  flux,  this  flux  consisting  of  a  ceaseless  inter- 
change of  energy  between  its  various  parts,  but  its  action 
is  always  uniform  and  unique.  It  is  this  in  variableness  and 
uniformity  which  renders  possible  a  philosophy  of  nature. 
The  investigation  of  the  manner  or  laws  under  which 
changes  take  place  is  comprised  under  such  names  as 
Physics,  Dynamics,  Mechanics,  or  Natural  Philosophy, 
but  since  in  this  book  we  shall  not  confine  ourselves  merely 
to  the  investigation  of  such  processes,  but  shall  consider 
ourselves  free  to  speculate  upon  them,  insofar  as  that  is 
possible,  the  latter  term  is  preferable.  It  was  used  by 
Newton  in  this  sense. 

The  ultimate,  or  primordial  causes  of  nature  must  ever 
remain  unknown  to  us,  since  the  infinite  cannot  be  grasped 
by  finite  minds,  but  a  certain  proximate  region  has  been 
explored  and  it  is  to  the  credit  of  mankind  that  its  borders 
are  being  continually  extended.  No  important  result 
can  ever  be  achieved  without  serious  and  concentrated 
thought  and  close  and  careful  reasoning.  Such  a  method,  in 
any  of  its  forms,  constitutes  mathematics,  and  any  man 
who  reasons  carefully  and  accurately  is  a  mathematician. 
The  theory  of  changes  was  called  by  Newton,  its  dis- 
coverer. Fluxions,  symbolizing  thereby  the  fluxions  of 
nature.  It  is  now  known  as  The  Calculus,  and  it  is  only 
decent  that  every  man  with  more  than  a  rudimentary 
education  should  know  the  calculus  and  thereby  some- 
thing of  the  universe  in  which  he  lives.  It  is  the  ability 
and  willingness  to  think  which  has  raised  civilized  man 


TZ>^'  i  '-f  ■  i «-, 


vi  PREFACE 

from  the  savage,  and  it  is  our  knowledge  of  nature  and 
natural  forces  which  determines  our  present,  or  any- 
future  civilization. 

The  study  of  such  a  subject  must  react  upon  the  student 
in  forming  within  him  a  new  realization  of  what  truth 
actually  is.  He  will  find  that  many  of  his  most  cherished 
beliefs  would  not  stand  before  the  Court  of  Nature,  or 
for  that  matter  before  an  ordinary  law  court.  For  ages  man 
has  seemed  incapable  of  distinguishing  between  the  true 
and  false,  and  has  had  little  desire  to  do  so  even  when  the 
means  were  at  hand.  The  idea  prevailed,  and  still  prevails, 
that  truths  could  be  created  by  authority.  A  statement  if 
loudly  proclaimed  and  accepted  by  a  sufficient  following 
was,  and  is,  supposed  to  be  true  irrespective  of  any  in- 
herent quality.  A  man  may  say  that  he  believes  and  ac- 
cepts as  a  truth  something  which  he  does  not  understand, 
but  unless  he  clearly  recognizes  for  himself  the  reasons 
why  a  thing  must  be  true,  it  is  not  a  truth  for  him.  The 
student  of  mathematics  learns  from  the  beginning  that 
nothing  but  the  truth  can  ultimately  prevail  and  that  what 
is  false  necessarily  carries  with  itself  its  own  annihilation. 
Authority  has  no  place  in  Science,  for  its  results  do  not 
rest,  or  need  to  rest,  upon  any  personal  sponsor,  no  matter 
how  distinguished,  but  solely  upon  the  truth  or  falsity  of 
the  reasoning  by  which  they  are  derived. 

Text  books  on  this  subject  have  been  overfull  of  prob- 
lems concerning  rods  and  strings  and  flies  walking  on 
circular  wires  or  perfectly  smooth  tables,  so  that  it  is 
perhaps  natural  that  the  impression  has  prevailed  that 
instead  of  being  an  instrument  for  the  acquisition  of 
truth,  mathematics  are  chiefly  an  agglomeration  of  symbols 
by  which  fantastic  results,  having  no  human  or  practical 
interest,  are  obtained.  But  natural  philosophy  is  not  the 
study  of  rods  and  strings.  It  is  the  story  of  the  universe. 
For  this  reason  we  shall  confine  ourselves  to  processes 
which  are  actually  occurring  about  us  all  the  time  —  to  our 
earth,  and  to  our  solar  system. 


PRINCIPLES  OF  NATURAL  PHILOSOPHY 


NATURAL  PHILOSOPHY 

1.     The  Fundamental  Law 

The  first  efforts  of  man  to  understand  the  universe  were 
by  metaphysical  processes.  Without  examining  the  uni- 
verse as  it  is,  they  sought  to  evolve  its  plan  out  of  their 
inner  consciences.  They  conceived  in  their  minds  a  uni- 
verse as  they  thought  it  might  be,  or  could  be,  or  should  be, 
and  sometimes  by  deduction  sought  to  fit  their  theory  to 
the  case.  Generally  they  did  not  trouble  themselves  to  go 
so  far  as  this,  but  rested  content  with  their  imaginary 
universe.    The    method    consisted    of    assuming    causes 
instead    of   interrogating    nature    herself.    They   further 
exhibited  the  strange  tendency  of  mankind  to  imagine  that 
by    the    use    of    sufficient    meaningless    words,    definite 
thoughts  could  be  expressed.  There  were  necessarily  as 
many  systems  as  philosophers.  But  truth  is  unique,  so 
that  only  one  of  all  these  systems  could  possibly  be  true, 
while  the  a  priori  probability  was  that  all  were  false. 
In  the  middle  ages  when  there  was  no  science  (knowledge) 
it  was  natural  that  men  should  have  exercised  their  minds 
with  "Beating  the  air,"  but  that  such  methods  should 
persist  to  the  present  day  is  an  anachronism.  Sooner  or 
later  these  relics  of  mediaevalism  must  disappear.  Among 
certain  workers  on  the  present  borderland  there  is  a  tend- 
ency to  revert  to  these  unsound   methods,   and  within 
recent  years  a  new  kind  of  metaphysical  physics  has  been 
developed.    The    student    of    natural    philosophy    must 
carefully  avoid  the  unclean  thing. 

We  recognize  that  the  universe  consists  of  matter 
although  we  do  not  know  what  matter  is,  or  when  or  how 
it  came  into  existence,  or  that  it  did  come  into  existence. 

1 


2  NATURAL  PKILOSOPHY 

We  simply  know  that  it  is  there.  We  further  recognize  two 
general  forms  of  matter  —  ordinary  gross  matter  and 
the  ether.  Gross  matter  is  of  various  kinds  —  some  70  odd 
elements  —  while  the  ether  is  apparently  uniform,  ex- 
tending through  all  space,  and  in  fact  occupying  all  space 
with  the  exception  of  that  occupied  by  the  atoms  of 
gross  matter.  We  further  recognize  that  none  of  this 
matter  is  ever  at  rest,  but  that  it  changes  its  position  rela- 
tively to  space  incessantly  —  or  it  is  always  in  motion. 
We  recognize  therefore  in  the  universe,  matter  and  motion, 
and  these  are  the  only  elements  of  which  we  have  any 
cognizance. 

The  ether  is  the  connecting  medium  of  the  universe, 
through  and  by  means  of  which,  gross  matter  imparts  its 
motion  to  other  particles  at  a  distance.  We  shall  see  that 
matter  in  motion  results  in  a  conception  called  force, 
and  also  in  a  conception  called  energy,  and  that  these  three 
inseparable  entities  —  motion,  force,  and  energy  —  are 
transferred  to  distant  points  through  the  medium  of  the 
ether.  If  two  particles  of  matter  were  separated  by  an 
absolute  vacuum,  i.e.,  a  space  containing  no  matter  of 
any  kind,  it  would  be  impossible  for  the  motion  of  one 
particle  to  impress  itself  upon  another,  or  to  influence  it 
in  any  way.  It  is  further  evident  that,  since  matter  is 
neither  created  nor  destroyed,  and  since  motion  in  a  body 
only  arises  after  it  has  been  transferred  to  it  from  some 
other  body  —  the  body  imparting  the  motion  losing  as 
much  as  it  transfers  —  the  total  amount  of  matter  and 
motion  in  the  universe  must  always  remain  constant. 

We  know  very  little  of  the  ether  except  that  it  is  a  fluid 
of  extraordinary  tenuity  and  under  a  very  high  pressure. 
What  its  density  and  pressure  are,  we  know  only  roughly, 
but  we  know  that  it  is  a  kind  of  matter,  since  it  possesses 
the  fundamental  property  of  all  matter,  viz.,  inertia. 
Inertia  means  literally  the  helplessness  of  matter,  or  the 
inabiUty  of  matter,  from  any  virtue  within  itself,  to  ac- 
quire motion,  or  when  in  motion  to  bring  itself  to  rest. 


THE  FUNDAMENTAL  LAW  3 

Some  external  'motion  or  force  is  necessary  to  produce 
motion  or  a  change  of  motion  in  matter.  The  pressure  of 
the  ether,  which  is  a  force,  must  be  due  to  some  kind  of 
internal  motion  in  the  ether,  but  of  the  nature  of  this 
motion  there  is,  as  yet,  hardly  a  surmise.  The  atoms  of 
gross  matter  take  up  and  reflect  this  motion  of  the  ether 
and  thus  become  points  or  centres  from  which  disturbances 
radiate  in  all  directions.  Every  atom  therefore  influences 
every  other  atom  in  the  universe,  and  the  observed 
attractional  and  repulsional  effects  are  due  to  these  radia- 
tions.* The  reservoir  of  motion  (energy)  is  therefore  the 
ether,  of  which  it  contains  an  infinite  amount.  It  has  been 
surmised  that  the  ether  is  the  ground  stuff  out  of  which  all 
gross  matter  has  been  formed  by  compression  —  the 
density  of  an  element  varying  with  the  conditions  of 
pressure,  and  possibly  temperature,  under  which  it  was 
formed.  But  of  such  matters  we  know  little  or  nothing. 
It  has  also  been  suggested  that  inertia  is  a  property  of  the 
ether  alone  and  that  gross  matter  in  an  ether  vacuum 
would  yield  to  a  push  without  any  resistance  and  cease  its 
motion  instantly  when  the  push  was  removed. 

Any  motion  of  a  body  necessarily  sets  up  a  disturbance 
in  the  ether  and  it  has  been  suggested  that  the  reaction 
against  the  body  of  such  a  disturbance  might  account  for 
the  resistance  to  motion  which  we  call  inertia.  But  granting 
such  a  possibility,  we  have  no  explanation  of  the  inertia  of 
the  ether.  Inertia  therefore  is  at  present  the  one  great 
mystery  of  nature.  It  is  not  enough  to  say  that  it  is  an 
inherent  property  of  all  matter,  for  there  must  be  some 
cause.  To  move  a  large  body  —  isolated  in  space  —  re- 
quires a  very  considerable  force  and  it  yields  to  this  force 
only  slowly  and  reluctantly,  but  when  once  in  motion 
there  is  an  equal  difficulty  in  bringing  it  to  rest. 

Experiment  shows  that  the  resistance  to  motion  of  all 

*The  author  has  shown  with  some  probability,  that  attraction 
and  repulsion  are  due  to  longitudinal  waves  in  the  ether,  v.  "Mechan- 
ics of  Electricity." 


4  NATURAL  PHILOSOPHY 

matter  is  proportional  jointly  to  the  mass,  or  quantity  of 
matter,  and  to  the  acceleration,  or  rate  at  which  the 
motion  changes.  Naturally  two  identical  bodies  would 
offer  twice  the  resistance  of  a  single  body,  and  when  no 
forces  are  applied  we  should  expect  the  state  of  motion,  or 
velocity,  to  remain  unaltered  both  in  amount  and  direc- 
tion. It  is  only  when  the  state  of  motion  is  being  altered 
that  we  should  expect  to  meet  with  any  resistance,  and 
such  we  find  to  be  a  fact. 

The  simplest  relation  between  change  of  motion  and 
resistance  to  motion  is  one  of  simple  proportionality  and 
experiment  proves  that  this  is  the  relation,  or  the  resistance 
is  proportional  to  the  rate  at  which  the  motion  changes. 
If  we  denote  the  resistance  by  r,  this  is  expressed  by  the 

dv 
relation,  r  =  —m-r-     We  define  a  force  as  the  product  of  a 

mass  by  its  acceleration.  To  change  the  motion  of  a  body, 

therefore,  we  shall  have  to  apply  a  force,  /,  equal  and 

dv 
opposite  to   r,    and    we   have   the   equation  /=wt- (1), 

dv 
where/  and  -r-  are  directed  quantities,  or  vectors,  having 

the  same  direction.    Equa.  (1)  is  Newton's  second  law  of 

motion  which  he  stated  "Change  of  motion  is  proportional 

to  the  applied  force  and  such  change  is  effected  in  the 

direction  of  the  force."  It  also  contains  his  third  law,  which 

is,  "The  action  of  the  force  is  equal  and  opposite  to  the 

reaction  (of  the  inertia)."  His  first  law  is  merely  a  corollary 

of  the  other  two. 

Equa.  (1)  may  be  called  the  Fundamental  Law  of  the 

universe,  for  from  it  we  shall  derive  all  the  principles  and 

laws  of  natural  philosophy.  There  is  nothing  in  it  which  is 

axiomatic  or  a  priori  evident,  although  Newton's  laws  have 

been  called  "Axiomata  Sive  Leges  Motus." 

dv 
But  the  fundamental  law,/  =  w  -r- ,  is  not  actually  a  law, 

or  it  does  not  express  exactly  the  relation  between  the 


THE  FUNDAMENTAL  LAW  5 

variables.  It  is  merely  the  statement  of  experimental 
results  under  certain  limited  conditions,  and  even  under 
these  conditions  the  law  gives  the  relations  only  to  a  very 
close  approximation.  If  a  force  acting  upon  a  body  at 
rest  in  the  air  imparts  a  certain  acceleration  in  unit  time, 
the  force  will  have  overcome  what  we  call  the  inertia  of 
the  body  and  in  addition  a  very  slight  resistance  from  the 
air,  provided  the  force  is  small.  Applying  the  force  continu- 
ously we  find  that  the  increment  of  the  acceleration  is  not 
constant,  but  is  a  function  of  the  previously  existing 
velocity.  A  curve  expressing  the  relation  between  the 
force  and  the  acceleration  will  be  nearly  a  straight  line 
at  the  beginning,  as  it  would  be  for  any  continuous  func- 
tion expressing  a  relation.  But  in  our  present  case  the  curve 
will  fall  away  because  the  resistance  of  the  air  increases 
with  the  velocity.  At  a  certain  velocity  the  resistance  is  as 
the  square  of  the  velocity,  and  when  the  body  has  a 
velocity  beyond  that  with  which  a  disturbance  travels 
in  the  air —  about  1100  ft.  per  second —  a  vacuum  forms 
behind  the  body,  because  the  body  moves  at  the  same  rate 
as  the  vacuum  tends  to  close  up.  A  constant  force  equal  to 
the  pressure  of  the  air  would  therefore  give  no  acceleration, 
but  merely  maintain  the  velocity  constant. 

Let  us  now  repeat  our  experiments  in  an  air  vacuum. 
We  now  find  that  the  curve  is  very  nearly  a  straight  line, 
or  the  resistance  of  the  inertia  is  very  nearly  proportional 
to  the  acceleration.  But  the  body  is  moving  in  the  ether 
and  any  motion  in  a  medium  is  resisted.  The  velocity  of  a 
disturbance  in  the  ether  is  about  186,000  miles  per  second, 
so  that  if  a  body  were  moving  with  a  greater  velocity  than 
this  it  would  require  a  force  equal  to  the  pressure  of  the 
ether  to  maintain  this  velocity  constant  without  imparting 
any  acceleration.  Now  the  greatest  velocity  in  gross  matter 
which  has  ever  been  observed  is  about  300  miles  per 
second.  This  is  insignificant  in  comparison  with  the  dis- 
turbance velocity  of  the  ether,  and  we  are  therefore  justi- 
fied in  concluding  that  our  law  —  verified  by  experiment — 


6  NATURAL  PHILOSOPHY 

holds  to  an  extreme  degree  of  approximation  for  all  matter 
moving  with  ordinary  velocities.  But  we  cannot  apply  the 
formula  to  matter  moving  with  extraordinary  velocities. 
There  are  certain  bodies  called  electrons,  which  are 
possibly  ethereal  vortical  rings  of  the  smallest  size  pos- 
sible (without  a  lumen),  which  may  move  with  a  velocity 
approaching  an  ether  wave.  As  such  bodies  meet  with  a 
resistance  which  increases  with  the  velocity,  it  has  been 
assumed  that  their  mass  increased  with  the  velocity,  and 
it  has  been  lightly  stated  that  the  mass,  or  quantity  of 
matter,  in  all  bodies —  gross  or  ethereal —  is  not  constant, 
but  a  function  of  the  velocity!  This  is  an  example  of  latter 
day  metaphysical  physics. 

The  amount  of  matter  in  a  body  is  at  all  times,  of 
course,  definite,  and  can  only  change  by  the  addition  or 
subtraction  of  matter. 

The  energy  of  a  body  in  motion,  called  its  Kinetic 
Energy,  is  defined  as  half  its  mass  into  the  square  of  its 

velocity,  or  7  =  -pr-  .  The  work  done  by  a  constant  force  is 

defined  as  the  product  of  the  force  into  the  distance  through 
which  it  acts  in  the  direction  of  the  force,  or  W  =fs. 
If  the  force  is  not  constant, 

J  2  n      dv  ,  n  w  /  2        2  \ 

i^^  =  J  ^d'r^  ^  J  ^^^^  ^  2  V2  ~  ^  / 
or  the  increase  of  the  kinetic  energy  is  equal  to  the  work 
done  upon  the  body  between  any  two  positions.  Kinetic 
energy  and  work  are  thus  equivalent,  and  we  may  define 
energy  in  general  as  that  which  is  capable  of  doing  work, 
and  the  doing  of  work  as  the  transference  of  energy  from 
one  body  to  another.  All  such  processes  are  evidently 
reversible. 

In  measuring  these  quantities  certain  standard  units  are 
selected,  usually  the  gramme  for  mass,  the  centimeter  for 
length  and  the  second  for  time,  these  units  constituting  the 
C.  G.  S.  system.  Accordingly  the  unit  of  force,  called  a 
dyne,  is  one  which  acting  upon  a  gramme  for  a  second 


W 


CENTRIFUGAL  FORCE 


imparts  to  it  a  velocity  of  a  centimeter  a  second.  The 

earth's  attraction  upon  a  gramme  at  its  surface  gives  it  a 

velocity  of  about  981  cms.  per  second,  for  every  second 

that  it  acts,  and  the  force  is  therefore  981  dynes.  The 

attractional  force  on  a  kilogram  is  1000  times  as  much, 

but  the  acceleration  is  the  same.  Hence  all  bodies  fall  (in  a 

vacuum)  in  the  same  time.  The  symbol  g  is  used  to  express 

this  constant  acceleration  at  the  earth's  surface. 

-.04  cms.     ,  dv  f.o4  J 

g  =  981 .    f  =  m-j-=mg=m  981  dynes, 

sec.  at 

where  m  is  expressed  in  grammes.  If  the  earth  and  moon 

were  brought  to  rest  and  allowed  to  fall  to  the  sun  from 

the  same  position,  the  sun  being  fixed,  they  would  perform 

the  journey  together  and  reach  the  sun  at  the  same 

instant.  The  accelerations  and  hence 

the  velocities  would  be  the  same  at 

all  times. 


2.     Centrifugal  Force 

Let  us  suppose  a  particle  of  mass, 
m,  attached  to  a  string,  OP,  and 
moving  in  a  circle  about  0  without 
gravity.  In  the  short  time  dt,  it 
would,  if  free,  move  to  A.  Calling 
/  the  average  pull  of  the  string,  both 
in  amount  and  direction,  we  have 


Fig.  1. 


AB  =/ 


2m 


d<p 


vstn-^z-.dt,  where  d(p  is  the  angle  POB,  and 


V  is  the  velocity  at  P.  Developing  the  sine  into  its  series, 

2mv 

We  shall  generally  use  the  Newtonian  notation  of  a  super-dot 
for  a  velocity  and  a  double  super-dot  for  an  acceleration.  At 


/d<p      d<p^  \  /        d<p^  \ 

It  ~  "48"+  ^^^')  ^  W2;^l  1  —  24    +  ^tc.  I- 


the  limit  where  d(p  becomes  zero,/  =  mv<p 


mv^ 


=  mp<f^. 


where  p  is  the  radius  of  curvature  of  the  path  at  any 
instant.  Hence  whenever  a  body  is  moving  in  a  curved 


8  NATURAL  PHILOSOPHY 

path  its  inertia  gives  rise  to  a  centrifugal  force,  or  force 
directed  away  from  the  centre  of  curvature,  which  is  equal 
to  its  mass  into  the  square  of  its  velocity,  divided  by  the 
radius  of  curvature.  This  force  measures  the  tension  of  the 
string  in  our  example.  Whenever  a  body  is  moving  freely 
in  a  curved  path,  as  a  planet  or  a  projectile,  the  component 
of  the  force  acting  upon  it  which  is  ±  to  the  path  must 
always  be  equal  to  the  centrifugal  force  and  opposite  to  it. 
This  component  obviously  cannot  influence  the  velocity 
of  the  body  but  merely  deflects  its  course.  A  body  thrown 
into  the  air  describes  a  parabola,  and  the  equation  of  the 
curve  referred  to  its  vertex  is  4:ay  =  x^,  where  a  is  the 
distance  of  the  vertex  either  from  the  focus  or  from  the 
directrix.  The  time  of  falling  from  the  vertex  to  the  level  of 

the  focus  is  t  =  \  ■ —     x  at  this  level  is  2a  and  the  constant 


horizontal  velocity  is  V  2ag.  At  the  vertex  the  centrifugal 

2  fH  a  2. 

force  is  therefore  • —  This  must  be  equal  to  mg  and  the 

P 

radius  of  curvature  at  the  vertex  is  2a. 

3.     Principle  of  Least  Action 

We  shall  now  examine  the  manner  in  which  motions  are 
executed  in  nature  with  a  view  to  ascertaining  whether 
any  particular  economies  are  effected.  When  a  system  is 
contained  in  a  closed  surface  across  which  forces  (motions) 
do  not  pass,  the  total  energy  of  the  system  remains  con- 
stant and  the  processes  consist  only  of  reversible  inter- 
changes between  work  and  kinetic  energy  within  the  sys- 
tem. Such  a  system  is  said  to  be  conservative  since  its 
total  energy  remains  constant.  But  if  during  these  changes 
the  work  is  not  transformed  wholly  into  gross  motion,  or 
gross  kinetic  energy,  but  some  part  of  it  is  expended  in 
producing  fine  (molecular)  vibrations,  such  as  heat 
(friction) ,  which  fine  motion  is  transferred  to  the  ether  and 
thus  lost  to  the  system,  then  the  system  is  non-conserva- 
tive and  its  energy  does  not  remain  constant. 


PRINCIPLE  OF  LEAST  ACTION  9 

In  a  conservative  system  the  kinetic  energy  of  any 
configuration  obviously  depends  simply  upon  the  position 
or  co-ordinates  of  that  configuration,  so  that  no  matter 
what  changes  it  has  undergone,  on  returning  to  the  same 
configuration  it  always  has  the  same  kinetic  energy. 
Hence,  taking  account  of  friction,  the  impossibility  of 
perpetual  motion  in  an  isolated  system. 

Let  us  suppose  that  a  system  changes  from  a  certain 
configuration  at  the  time  h,  by  its  own  free  motion,  to 
another  configuration  at  the  time  ^2,  the  coordinates  of 
any  one  particle  of  the  system  at  any  instant  being  x,  y,  z. 
If  we  vary  the  motion  by  guides  or  constraints  so  that  at 
any  instant  a  particle  instead  of  occupying  its  natural 
position  X,  y,  z,  occupies  a  position,  x  -\-  8x,  y  -^  8y,  z  -{-  8z, 
where  8x,  8y,  8z  are  infinitesimal  arbitrary  quantities, 
but  the  initial  and  final  positions  are  the  same,  the  differ- 
ence between  the  work  done  in  the  free  paths  and  in 
the  varied  paths  is 

8W  =  1:  {X8x  +  Y8y  +  Z8z)  = 

d  (dy  ^\  ,  d  (dz  .  \       (dx    ,  8x   .  dy   ,  8y  .  dz    ,  8z\l 

where  X,  Y,  Z  are  the  components  of  the  force  acting  upon 
any  particle  and  t  is  the  independent  variable. 

The  last  term  is  the  variation  of  T,  the  kinetic  energy, 

-  -=-f[(sy+(i;+(i:)> 

Integrating,      8    C  (w  +  T\dt  =  Sm/'^Sx  +  ^^8y  + 

dt^y\h 


10  NATURAL   PHILOSOPHY 

Since  the  variations  vanish  at  the  upper  and  lower  limits, 
'{W  +  T)dt  =  0. 


\r 


Negative  work,  or  stored  up  energy,  is  called  Potential 
energy,  designated  by  V,  and  W  =  —  V. 

Hence  8    f  V  -  V)dt  =  0. 


,r<- 


This  result  is  known  as  Hamilton's  Principle.  It  is  entirely 
general  and  applies  to  non-conservative  as  well  as  con- 
servative systems.  Since  in  a  conservative  system  51^  =  8T, 

d    rV  +  W)dt  =  25    r  Tdt  =  0. 

For  the  natural  paths,  therefore,  these  integrals  have 
a  stationary  value,  or  for  all  other  infinitely  near  paths 
they  are  either  greater  or  less.  As  a  matter  of  fact  all  these 

integrals  are  minima,  for  taking  the  expression  5    i    Tdt  =  0, 

it  is  evident  that  by  causing  a  particle  to  execute  an  in- 
finitesimal loop  at  any  point  of  the  actual  path,  the  in- 
tegral must  be  greater  than  for  the  actual  path,  and  the 
actual  or  free  path  cannot  be  a  maximum. 

The  integral    I    Tdt  is  called  the  action  of  the  system, 

and  we  have  proved  that  in  a  conservative  system  the 
natural  action  is  less  than  that  in  any  other  infinitely 
near  path.  This  is  known  as  the  Principle  of  Least  Action. 
It  means  that  the  average  work,  or  kinetic  energy,  for  the 
time,  or  the  time  mean  of  the  work  or  kinetic  energy,  mul- 
tiplied by  the  time,  cannot  possibly  be  made  less.  Or  if 
we  desire  to  effect  a  given  change  in  any  way  different 
from  the  natural  way,  we  shall  have  to  expend  more 
energy,  or  do  more  work  than  nature  does,  provided  the 
change  is  effected  in  the  same  time. 

This  is  a  remarkable  result  contained  implicitly  in  the 
fundamental  law. 

Let  us  take  the  case  of  a  body  moving  under  the  in- 


PRINCIPLE   OF   LEAST   ACTION 


11 


fluence  of  gravity  between  any  two  points.  Taking  the 
vertex  of  the  path  as  origin,  the  time  as  abscissas  and  the 
work  as  ordinates,  the  action  between  0  and  2  will  be  rep- 

2  ' 
072  will  be  a  parabola,  though 
bola  of  the  path.    The  action, 


resented  by  the  area  023.  Since  W  =  ^'  the  action  curve 

not  of  course  the  para- 
0   6       3 


s 


o 


Witi 


.  The  time 


mean  of  the  work,  or  the  average 
work  for  the  time  is  therefore 
}4  of  the  total  work,  and  the 
product  of  this  mean  by  the 
time,  or  the  area  023  =  0453  = 
the  action,  cannot  possibly  be 
made  less.  If  the  motion  is  be- 
tween two  points  on  the  same 
level,  corresponding  to  1  and  2, 
the  action  is  zero,  for  the  action  between  1  and  0  is  a  minus 
area,  since  W  is  negative  in  this  part,  and  the  action  between 
0  and  2  is  an  equal  positive  area.  There  are  two  free  paths 
by  which  a  body  may  move  between  any  two  points —  say 
points  corresponding  to  1  and  7  —  but  the  action  in  each 
case  will  be  the  same,  viz.,  076  —  023  and  —6723.  Between 
points  corresponding  to  1  and  2  there  are  two  free  paths 
and  an  infinite  number  of  possible  guided  paths,  but  in 
every  case  the  action  is  zero.  In  this  case  alone,  which  is  an 
absolute  minimum,  it  is  possible  for  a  guided  path  to  have 
no  greater  action  than  the  free  path,  but  in  all  other  cases 
the  free  action  is  less  than  any  constrained  action,  and 
in  all  cases  the  free  action  is  the  least  possible.  Nature, 
therefore,  takes  no  care  of  time  but  is  very  exacting  as  to 
how  her  energy  shall  be  expended.  She  insists  that  the 
action  shall  be  the  least  possible.  When,  however,  no 
energy  is  expended  she  becomes  economical  both  as  to 


time  and  space 


'.r™-?g 


vds  =  0.  When  no  energy 


12  NATURAL   PHILOSOPHY 

is  expended,  v  is  constant,  and  s  is  a  minimum  as  well  as  t, 
for  the  path  between  any  two  points  becomes  a  straight 
line.  A  body  under  no  forces,  or  a  ray  of  light,  moves  in  a 
straight  line,  or  they  traverse  the  shortest  possible  path 
between  any  two  points  in  the  shortest  possible  time. 

4.    Brachystochrone  and  Tautochrone 

Whenever  energy  is  expended  it  is  easy  to  improve  upon 
nature  in  the  matter  of  time,  and  we  shall  construct  a  path 

such  that  a  body  moving 
under  gravity  may  pass  from 
one  point  to  another  in  the 
least  possible  time. 
-^1  ^-"^^  ^  i^  ^^  .B  Let  us  suppose  (Fig.  3) 
f\  [        \         J        that  our  body  is  at  the  point 

\^^V     \<>^)jt^  1  a-nd  we  wish  to  conduct  it 

^  to  the  point  2  in  the  shortest 

p       ^  possible  time.  It  has  a  veloc- 

ity 2^0  at  1,  and  we  take  as 

our  base  of  ordinates  a  line  AB  which  is  ^r  above  1. 

2g 


t=    I    —  =    I   = — dy,  where  X  =--T- 

■J 


8dx  _  dx.  x' 

r  6xd.  (—  ^'      \ 

J  VVI  -\-x''-  yl2gy/ 

The  variations  vanish  at  the  points  1  and  2,  so  that  for 

^  to  be  a  minimum,  the  last  term  must  vanish,  or 

x'  1 

■=  must  be  a  constant  —  say =>  where 


^\+x'^  V  2gy  2yjag 

a  is  a  constant  to  be  determined  directly.  If  r  is  the  angle 

which  the  path  at  any  point  makes  with  the  y  ordinate, 

x' 
sin  r  = —  .     Hence,    the   required    path,    in   terms 

of  y  and  r,  is  2a  sin^r  =  y.  This  is  the  equation  of  a  cycloid 


BRACHYSTOCHRONE   AND   TAUTOCHRONE  13 

where  a  is  the  radius  of  the  generating  circle  and  y  is 
measured  from  its  base  Hne.  The  solution  is  entirely- 
general  and  unique,  since  a  circle  can  always  be  found 
which  rolling  along  a  given  base  line  will  trace  with  one  of 
its  points  a  curve  which  shall  pass  through  two  given 
points,  and  there  is  only  one  such  circle.  The  radius  of  the 
generating  circle  is  readily  found  from  the  data. 

The  cyloid  has  also  the  property  that  the  time  taken  in 
falling  from  rest  to  the  lowest  point,  C,  of  the  curve,  is  the 

ds  V  1  4-  x'^ 

same  for  all  starting  points.  For  dt  =  —  =  dy, 

V         ^2g{y-y,) 

where  y\  is  the  ordinate  of  the  starting  point. 

t^      {''jA±^dy, 
yiJ      ^2g{y  -  yi) 


But  l+x'^  =  ,r^^^—,  and  t  =      ('""J ^ = 

2^  -  y  yj      llgiy  -  y^)  {la  -  y) 

2^    sin  "Vf^^r  =  -V^- 
J  g  J2a  -  y{[yi  ^g 

Or  the  time  is  independent  of  the  starting  point. 
The  time  of  a  complete   oscillation  is  2x  ^— .    Since  a 

cycloid  is  the  involute  of  two  equal  cycloids  placed 
above  it  as  in  Fig.  3,  or  since  it  can  be  described  by 
fixing  a  string  of  length  4a  at  D  and  wrapping  it 
around  the  upper  cycloids,  it  is  evident  that  small  os- 
cillations of  a  particle  about  the  lowest  point,  C,  will  co- 
incide very  nearly  with  those  of  an  ordinary  simple 
pendulum  of  length  4a.  Hence  the  time  of  a  small  oscilla- 
tion of  a  simple  pendulum  is  2^-^—,  I  being  its  length. 

We  can  easily  derive  this  tautochronous  property  of  the 
cycloid  in  another  way.  The  intrinsic  equation  of  a  cycloid 
is  5  =  4a  sin  <p,  where  s  is  the  length  of  the  curve  measured 
from  any  point  and  <p  is  the  angle  between  the  directions  of 
the  curve  at  the  first  and  any  second  point.  Measuring  5 
from  the  lowest  point,  since  the  acceleration  along  the 


14 


NATURAL  PHILOSOPHY 
4a 


curve  IS  5  =  —  g  sm  ^,  5  = 


g 


5  (1).  Such  a  motion  where 


the  force  of  restitution  is  proportional  to  the  distance  from 
the  position  of  equilibrium  is  called  harmonic  motion,  and 
a  particle  oscillating  about  the  lowest  point  of  a  cycloid 
therefore  executes  a  harmonic  motion.  Integrating  (1), 


-VJ 


(5i2—  s2),  where  si  is  the  starting  point. 


t  = 


-1 


'4a     •    "^  ^ 
^"  sm     — 

g  ^1 


^1 


and  the  complete  period  is 


=  27r^_,  which  is  independent  of  the  starting  point. 


We  shall  verify  the  principle  of  least  action  by  deter- 
mining directly  what  the  path  of  a  body  moving  under 
gravity  must  be  in  order  that  the  action 
between  any  two  points  of  the  curve 
shall  be  a  minimum.  The  body  is  pro- 
jected from  1  with  a  velocity  Vq  and 
guided  (if  necessary)  along  a  curve  pass- 
ing through  2,  such  that  the  action  be- 
tween these  two  points  is  a  minimum. 
Take  as  origin  of  y  co-ordinates  a  base 


Fig.  4. 


line  OX,  such  that  yi  =-— 

2g 


Then  r'^dt=~  ^vds—  P  ^2gy  ^l+x'2.dy, 

and  this  integral  must  be  a  minimum.   Considering  y  the 
independent  variable  and  varying  x,  we  have 

5   f"  V2^  Vl  -]-x%dy=    C^—iM=x'ddx=0=^ 
ij  I J      Vl-t-r^'2 


x'8x  V2g:vP 


Vl  +x 
For  the  action  to  be  a  minimum, 


'2|l  ij  \    -Jl+X'2        J 

^2gy 


^ll+x'2 


x'  must  be  a  con- 


CENTRAL   FORCES  15 

stant,  c.  When  T  =  ;r"  >  "the  curve  is  horizontal  and  it  does 
2g  _ 

not  extend  above  this  level  since  values  become  imaginary. 
Let  c2  =  lag,  where  a  will  be  determined  directly.  Hence 

x'  =  -^    ^     and  ^  =  2  Va   ^y  —  a  +  C.  We  have  not  yet 
jy  —  a 

fixed  our  origin  of  co-ordinates,  but  shall  take  it,  so  that 

C  =  0.      Hence   the  required   path  is   x2  =  Aaiy  —  a). 

This  is  the  equation  of  a  vertical  parabola  with  its  directrix 

as  the  X  axis,  and  a  the  distance  of  the  vertex  from  the 

directrix,  or  from  the  focus.  The  velocity  of  any  projectile 

at  its  highest  point  is  thus  the  same  as  if  it  had  fallen 

to  that  point  from  the  directrix  from  rest.  The  required 

path   of   least   action   is   therefore   a    parabola    passing 

through  the  two  points  and  bearing  a  certain  relation  to 

the  initial  velocity.  Since  three  conditions  fix  any  conic, 

the  parabola  is  determined.  But  this  parabola  is  precisely 

the  free  path. 

6.  Central  Forces 

Let  a  particle  of  mass  m  move  subject  to  a  force  which 
radiates  from  a  fixed  point.  Let  r  be  the  distance  Jrom  the 
point  to  the  particle  at  any  instant  and  <p  the  angular 
velocity  of  this  radius.  The  forces  acting  along  a  radius  are 
the  centrifugal  force  and  the  force  /  radiating  from  the 
point.  The  effective  force  along  a  radius  is  mr,  and  mr  = 
nir<p2  +  /  (!)•  If/  attracts  the  particle  it  is  negative  and 
opposes  the  centrifugal  force.  Integrating  (1) 

ij  1  2    |i  2        1        ij 


mr2(pd(p 


•      2  ,  mr2<p2  \  2        ri 

=  mr2      -\ ^—  \    —    \    tp  d  {mr2<p).    The   last  integral 

1 1  ^      1  1     1 J 

must  therefore  be  zero,  and  mr2(p  =  mrvc  is  constant, 
where  Vc  is  the  circumferential  velocity,  or  the  velocity  ± 
to  the  radius.  It  will  be  noted  that  the  centrifugal  force, 
being  an  internal  force  arising  solely  from  inertia,  can  do 
no  work  upon  the  particle,  the  increase  or  decrease  of 


16  NATURAL   PHILOSOPHY 

kinetic  energy  being  due  solely  to  the  external  force,  /. 
The  product  of  a  directed  quantity  by  its  distance  from  an 
axis  is  called  the  moment  of  the  quantity  about  that  axis. 
The  moment  of  the  angular  velocity  is  evidently  the 
linear  circumferential  velocity.  The  moment  of  a  force 
about  an  axis  is  called  a  Couple.  If  two  equal  and  opposite 
forces  act  at  right  angles  to  the  extremities  of  a  line  which 
is  fixed  at  its  centre,  the  measure  of  such  a  couple  is  the 
product  of  one  of  the  forces  into  the  distance  between  them, 
or  it  is  the  moment  of  one  of  the  forces  about  the 
extremity  of  the  line  joining  the  two  forces.  The  product  of 
a  mass  by  its  velocity  is  called  the  momentum  of  the  mass, 
or  it  is  the  quantity  of  motion  in  a  body. 

We  have  obtained  the  important  principle  that  a  body, 
under  the  action  of  a  central  force  only,  preserves  the 
moment  of  its  momentum  about  the  centre  constant. 

The  area  described  by  a  radius  is  >^    I    r^dcp  =  Ct,  and  this 

area  is  proportional  to  the  time,  which  is  Kepler's  second 
law.  If  the  body  is  subject  also  to  circumferential  forces, 
it  does  not  preserve  its  moment  of  momentum  constant. 
For  the  moment  of  the  elementary  circumferential  force, 

or  the  elementary  couple,  is  m  -7-  (r^cp)  ='yyi  -rr  (^^c)>  and  the 


time  integral  of  this  is  mrvc 


We  have  thus  a  second  important  principle  —  the  time 
integral  of  a  couple  about  a  fixed  axis  is  measured  by  the 
increase  (or  decrease)  of  the  moment  of  momentum  about 
that  axis.  Or,  since  the  moment  of  momentum  of  a  body  is 
its  moment  of  inertia  about  an  axis  into  the  angular  velocity 
about  the  axis,  this  is  evidently  the  time  integral  of  the 
couple  acting  about  that  axis.  Further,  since  the  kinetic 
energy  about  any  axis  is  half  the  moment  of  inertia  about 
the  axis  into  the  square  of  the  angular  velocity,  this  is 
evidently  the  angle  integral  of  the  couple.  For  a  particle, 
w,  the  couple  at  any  instant  is  mrDtr<p,  and  the  angular 


ATMOSPHERIC   CIRCULATION  17 

2 


integral  is  m    I    r(pd{r<p)  =  m—Ty- 


,  or  the  angle  integral  of 
the  couple  is  measured  by  the  increase  of  the  kinetic  energy. 


7.  Atmospheric  Circulations 

The  following  problem  is  fundamental  in  the  theory  of 
the  circulation  of  our  atmosphere,  or  of  any  planetary 
circulation.  The  question  is,  what  path  would  a  moving 
mass  of  air  (wind)  describe  on  a  rotating  spheroid,  if 
such  mass  were  practically  unhindered  by  other  masses  of 
air,  or  by  friction. 

Let  us  suppose  a  plastic  mass  rotating  about  an  axis  and 
subject  to  its  own  gravitation.  If  it  were  not  rotating  it 
would  assume  a  strictly  spherical  form  and  the  lines  of 
force  at  the  surface  would  all  pass  through  the  centre. 
The  centrifugal  force  of  rotation  causes  the  mass  to  bulge 
at  its  equator  and  transforms  the  sphere  into  a  spheroid, 
the  lines  of  force  at  the  surface  being,  as  before,  ±  to  the 
surface.  If  co  is  the  angular  velocity  of  rotation  and  ^  the 
angle  which  a  radius  makes  with  the  equatorial  plane, 
the  centrifugal  force  at  its  extremity  is  r  cos  d^  co^,  and 
the  resolved  part  of  this  along  the  surface,  towards  the 
equator,  is  r  sin  ^  cos  t?  0)2  to  a  close  approximation,  if 
the  spheroid  differs  but  little  from  a  sphere.  For  a  particle 
to  be  in  equilibrium  on  the  surface,  the  resolved  part  of 
the  gravitational  force  along  the  surface,  towards  the  pole, 
must  be  exactly  equal  to  the  surface  centrifugal  component. 
This  gravitational  component  would  evidently  restore  the 
spheroid  to  its  original  spherical  form,  immediately  the 
rotation  ceased.  For  a  particle  to  be  in  equilibrium  at  any 
point  on  the  surface  it  must  therefore  have  exactly  the 
rotational   velocity,   co.    If   the   particle  has   an   angular 
velocity,  ^p,  about  the  axis,  the  force  urging  it  along  a 
meridian  is  /  =  r  sin  t?  cos  ^  {oo'^  —  yp'^  ) ,  o,  positive  value 
denoting  acceleration  towards  the  pole  and  a  negative 
value  towards  the  equator.  The  moment  of  momentum 
about  the  axis  must  remain  constant  since  there  is  no 


18  NATURAL   PHILOSOPHY 

couple  about  this  axis,  or  cos2  t?i/'  =  C.     Hence  f  =  r^  = 

r  sin  T?  cos  r?  ( co2 —  V    Integrating  and  putting  r>  =  t?o 


when  t?  becomes  zero,  we  have 
i>2        cos  2t?o  -  cos  2t? 


in  T?  cos  r?  [  co2 —  \     Integrating  and  putting  r>  =  t?( 

Lve 
^  .   aj2  +  O  (tan2  t?o  -  tan2  i^). 

cos  2t?o  -   cos  2i?  =  2  C  /"-^  -  ^  j 

and  tan2t?o  -  tan^i}  =  ^  ( lAo  -  ^Aj- 

Hence  t?2  =  C  (t/'o  -  ^)  A  -  ^  j  (1).      There  are  two 

values  of  ^  which  makes  t^  vanish.     One  of  these  we  have 

already  taken  as  xpo,  and  the  other  is  -i—     The  path  is 

tangent  to  a  parallel  of  latitude  at  these  points  and  the 
whole  curve  must  lie  between  these  two  parallels.  Writing 
the  angular  velocities  at  the  upper  and  lower  parallels  as 

ypu  and  ^l/o,  we  have  ^^  =  -^  {■^p^  _  ^)  (^  _  ^^)   (2).      We 

shall  call  linear  velocities  along  a  parallel,  horizontal  veloci- 
ties, and  linear  velocities  along  a  meridian,  polar  velocities, 

designated  by  vu  and  v^,.     Since  \l/  =  ^   =  r»  we 

^  r  cos  t?        cos2  ?? 

have  from  (2),  Vp2  =  i-  {vh^  -  Vo^)  {v^^  -  Vh^)  (3). 

It  must  be  borne  in  mind  that  these  are  absolute  hori- 
zontal velocities — not  relative  to  the  earth.  Designating  the 
relative  horizontal  velocity  by  Vrh,  Vrh  =  r  cos  t?  (co  —  ^)  = 

-rC         r^  (i)C  ... 

r  cos  t?  CO = vu  (4). 

cos  ?>  Vh 

Let  Vr  be  the  total  velocity  relative  to  the  earth.  Then, 
from  (3)  and  (4), 

■M^„,  ^±     0  r^o  0      o      •  /-        ^0  cos  t?o       "i^u  COS  T?„         J 

Now  n  0)2  62  =  vq^  Vu^,  smce  C  =  — =  — -,  and 


ATMOSPHERIC   CIRCULATION  19 

co2  =  \Po  xj/u.  Hence  Vr^  =  vq^  —  2r2coC  +  ^u^  (5).  Or  the 
velocity  relative  to  the  earth  is  constant.  It  will  be  seen 
that  the  maximum  polar  velocity  occurs  on  that  parallel 
where  there  is  no  poleward  acceleration,  or  where  \p  =  cc  = 
^\j/o  yj/u.  This  maximum  polar  velocity  is  v^  —  Vq.  Designat- 
ing the  absolute  horizontal  velocity  at  this  parallel 
of  equilibrium  by  Ve,  ^  =  v^  Vq,  or  the  horizontal  velocity 
at  this  point  is  the  geometric  mean  of  the  extreme  hori- 
zontal velocities.   Since  cos2  t?o^o  =  cos2  ^u'^u  and  ^i*  =  -7-. 

r  cos  ^Q-j/Q  =  r  cos  t?„co,  or  the  absolute  horizontal  velocity 
at  the  lower  limit  is  equal  to  the  velocity  of  the  earth 
at  the  upper  limit,  and  the  absolute  horizontal  velocity 
at  the  upper  limit  is  equal  to  the  velocity  of  the  earth 
at  the  lower  limit.  The  motion  is  thus  completely  deter- 
mined. 

As  on  all  planets  the  temperature  is  greater  on  the  whole 
at  the  equator  and  becomes  gradually  less  towards  the 
poles,  the  atmosphere  rises  at  the  equator  and  is  replaced 
by  other  portions  flowing  along  the  surface  from  north  and 
south.  Such  streams,  effected  by  differences  of  tempera- 
ture, will  be  forced  to  execute  such  paths  as  we  have  just 
determined,  and  as  in  crowds  when  a  general  trend  is  once 
established  there  is  little  mutual  interference,  so  the 
circulation  of  the  air  must  approximate  closely  to  the 
dynamical  factors. 

The  equatorial  circulation  of  a  planet  is  shown  in  Fig.  5. 
The  currents  flowing  towards  the  equator  are  at  first 
close  to  the  surface  but  are  continually  deflected  —  to  the 
right  in  the  northern  hemisphere,  to  the  left  in  the  southern 
hemisphere.  They  do  not  reach  the  equator,  but  are  de- 
flected due  west  at  a  high  level.  At  the  parallels  of  equilib- 
rium, indicated  by  the  dotted  lines,  the  directions  are 
due  north  and  south,  the  upper  currents  going  poleward 
while  the  surface  currents  are  towards  the  equator.  The 
limiting  parallels,  north  and  south,  are  functions  of  the 
difference  of  temperature  between  these  parallels  and  also 


20  NATURAL   PHILOSOPHY 

of  the  rotational  velocity  of  the  planet.  Since  the  curva- 
ture of  the  paths  is  a  minimuni  nearest  to  the  equator, 
the  general  trend  of  this  circulation  is  constantly  to  the 
west. 

There  is  likewise  a  fiat  polar  circulation  the  extent  and 
characteristics  of  which  are  determined  to  a  certain  extent 
by  the  elements  we  have  just  discussed.  Between  the  equa- 
torial and  polar  circulations  is  the  temperate  circulation 
composed  of  several  partly  independent  and  not  sharply 


Fig.  5. 

differentiated  zones.  The  temperate  circulation  as  a  whole 
moves  towards  the  east  with  varying  northerly  and 
southerly  components.  As  Lord  Kelvin  has  pointed  out, 
there  is  on  the  whole  a  slow  shifting,  due  to  friction,  of  the 
surface  currents  towards  the  poles  with  a  counterbalanc- 
ing slow  shifting  at  higher  levels  from  the  border  of  the 
polar  circulation  to  that  of  the  equatorial  circulation. 
The  circulation  of  any  planetary  atmosphere  is  thus 
differentiated  into  six  distinct  circulations,  the  borders  of 
which  are  very  sharply  marked.  The  currents  of  the  equa- 
torial circulation  are  very  constant  both  as  to  their  in- 
tensities and  the  shapes  of  their  paths,  the  polar  circulation 
less  so,  while  the  temperate  circulation  is  still  less  stable. 
It  is  hardly  necessary  to  state  that  what  would  be  a  con- 
stant and  stable  condition  in  all  the  circulations,  pro- 
vided our  postulated  conditions  existed,  viz.,  that  the 
earth  had  a  homogeneous  surface  and  were  symmetrically 
heated  about  the  equator,  does  not  actually  exist  because 


MOTION    OF   RIGID    MASSES  21 

the  inclination  of  the  sun  to  the  equator  is  constantly- 
shifting  and  because  the  surface  is  irregularly  divided  into 
land  and  water,  the  land  being  of  varying  altitudes.  In 
the  ideal  conditions  the  equatorial  circulations  would 
never  reach  the  equator,  while  in  the  actual  conditions 
they  frequently  cross  it.  This  leads  at  times  to  cyclones 
and  various  other  abnormal  disturbances,  a  fuller 
discussion  of  which  will  be  found  in  "The  Atmosphere," 
by  the  author.  It  is  interesting  to  note  that  Dr.  Percival 
Lowell  has  observed  ** Faint  lacings.  .  .  criss-crossed 
by  darker  lines"  in  the  equatorial  zones  of  both  Jupiter 
and  Saturn.  It  is  quite  possible  that  these  are  cloud 
streams  in  their  equatorial  circulations,  and  a  glance 
at  Fig.  5  shows  that  they  might  have  just  such  an 
appearance  when  viewed  in  a  telescope.* 

From  the  law  of  constant  moments  of  momentum, 
4/  =  2C sec2 ^  tan M  =  2yp  tan M,  and  RcoS'&^  =  2R^ sin t?^. 
Putting  ypr  for  the  relative  angular  horizontal  velocity, 
or  i/'y  =  ;/'  —  CO,  we  have  R  cos  ■&4^  =  2R  {xj/r  +  co)  sin  ^4, 
and  R^  =  —R  sin  i}  cos  ^.2o)-j/r  approximately,  if  xpr  is 
small  compared  with  co.  Hence  we  may  write  approxi- 
mately, R  cos  ^^  =  2Roi  sin  M.  Consequently  if  p  be 
the  radius  of  curvature  of  the  path  and  Vr  the  relative 

V  ^  1) 

velocity  at  any  point,—  =  2co  sin  i^Vr,   or  p  =  tj ^ — jr. 

p  zco  sm  V 

The  curvature  of  the  path  is  therefore  nearly  proportional 

to  the  sine  of  the  latitude  and  inversely  proportional  to  the 

relative  velocity.  This  result  was  first  given  by  Ferrel. 

8.     Motion  of  Rigid  Masses 

We  have  hitherto  considered  the  motion  of  particles,  or 
of  masses  of  matter  supposed  concentrated  into  a  mathe- 
matical point.  We  shall  now  investigate  the  motion  of 
masses  having  definite  dimensions.  We  can  consider  a  rigid 
body  as  made  up  of  an  infinite  number  of  particles  which 
are  held  together  by  an  unyielding  non-material  frame. 
*Dr.  Percival  Lowell.     Popular  Astronomy.     April,  1910. 


22  NATURAL   PHILOSOPHY 

In  any  field  of  force  every  particle  will  be  subjected  to  a 

force,  F,  which  we  shall  call  the  applied  force.  It  cannot 

obey  this  force,  as  a  detached  particle  would  do,  by  reason 

of  its  fixed  connections,  but  the  effective  force  on  each 

particle  will  be  the  geometrical  resultant  of  the  applied 

force  and  the  sum  of  the  reactions  of  the  neighboring 

particles,  and  the  particle  will  obey  this  effective  force  the 

same  as  if  it  were  free.  That  is,  the  actual  infinitesimal 

path,  ds,  will  be  in  the  direction  of  this  force,  and  it  will 

(Ps 
oppose  to  this  force  its  inertia,  measured  by  m-7— ,  which 

likewise  measures  the  effective  force.  Summing  all  the 
forces  we  have  three  groups  —  the  applied  forces,  the 
reactions  of  the  neighboring  particles,  and  the  forces  of 
inertia  or  the  effective  forces.  Now  the  sum  of  the  reactions 
among  the  particles  must  be  zero,  since  there  is  no  relative 
motion  between  them.  Hence  the  geometric  sum  of  all  the 
applied  forces  must  be  equal  to  the  geometric  sum  of  all 

the   effective  forces.    Or   SF  =  Sm-r—,    where  ds  is  the 

actual  elementary  path  of  each  particle,  and  S  signifies 
geometric  sum — not  algebraic  sum.  This  is  D'Alembert's 
Principle. 

Taking  any  ±  axes,  and  resolving  each  applied  force, 
F,    into    X,    y,    Z,    parallel    to    these    axes,    we    have 

Putting  l^mx  =  Mx,  Zmy  =  My,  Xmz  =  Mz,  where  M  is  the 
total  mass,  these  equations  determine  a  point  in  the  body 
having  co-ordinates  oc,  y,  2,  and  this  point  is  called  the  centre 
of  inertia,  or  centre  of  mass.  It  is  evidently  a  fixed  point 
in  the  body,  irrespective  of  whatsoever  forces  act  upon  it. 
The  interpretation  of  these  equations  is  that  if  we 
transfer  every  applied  force  to  the  centre  of  inertia,  parallel 
to  itself,  the  geometrical  sum  of  these  forces  will  be  equal 
to  a  single  force  acting  upon  the  entire  mass  considered  as 
concentrated  at  this  point  and  this  force  will,  of  course, 


MOTION   OF   RIGID    MASSES  23 

be  equal  and  opposite  to  the  inertianal  force  of  such  a 
concentrated  mass.  In  other  words,  the  motion  of  the 
centre  of  inertia  will  be  same  as  the  motion  of  a  ma- 
terial point  or  particle  of  mass  M,  under  a  force  which  is 
the  geometrical  resultant  of  all  the  applied  forces  acting 
at  this  point  parallel  to  their  original  directions. 

We  have  next  to  consider  that  the  applied  forces  do 
not  act  at  the  centre  of  inertia,  but  on  the  several  particles. 

Since  sy  =  Sm-^,  l^xY  =  Sm^c  -^  and  S^^X  =  ^my -^, 

xY  is,  the  moment  of  the  force  Y  about  the  axis  of  Z, 
and  yX  is  the  moment  of   X  about  this  axis.     Hence 

i:,{xY—  yX)  =  ^mfx-r^  —  y  -7^  j  means  that  the  couple 

about  the  z  axis  due  to  all  the  applied  forces  is  equal  and 
opposite  to  the  couple  about  this  axis  due  to  the  inertianal 
forces.  We  have  then, 

;(xmx\ 
h^myj 

and 

B.        .(,z-.y)  =  ..(.g-.S)  = 

d  ^    /    dz  dy\ 

_  s^^y  J,  -  ^  ^  j 

d  -,    /   dx  dz\ 


d2z  d2  /  \  J2 

dt2        dt2\  J       dt2      - 


24  NATURAL   PHILOSOPHY 

Equas.  A  state  that  the  sum  of  the  momenta  of  all  the 
particles  in  any  direction  is  equal  to  the  component  in 
that  direction  of  the  momentum  of  the  total  mass  moving 
with  the  velocity  of  the  centre  of  inertia. 

Equas.  B  state  that  the  derivative  with  respect  to  the 
time  of  the  moment  of  momentum  about  any  axis  is  equal 
to  the  couple  about  that  axis,  a  result  which  we  have 
already  obtained.  It  follows  that  the  motion  of  a  rigid 
body  under  the  action  of  any  forces  can  always  be  re- 
solved into  a  translational  motion  of  the  centre  of  inertia 
and  a  rotation  about  an  axis  through  that  centre.  It  is 
further  evident  that  these  two  motions  are  entirely  in- 
dependent of  each  other,  so  that  if  we  oppose  the  transla- 
tional motion,  the  rotation  will  occur  as  before,  and  if  we 
prevent  the  rotation  the  translational  motion  will  be  un- 
influenced. 

We  can  arrive  at  these  results  more  simply  as  follows: 
The  applied  forces  can  be  reduced  (geometrically)  to  a 
single  force  acting  through  some  line  within  or  without 
the  body  (but  if  without  the  body  to  be  considered  as 
rigidly  connected  with  it) ,  and  the  geometric  sum  of  all  the 
inertianal  forces  is  a  single  force  acting  through  this  same 
line,  but  in  the  opposite  direction.  Dropping  a  ±  from  the 
centre  of  inertia  to  this  line,  and  applying  to  the  centre  of 
inertia  a  force  equal  and  parallel  to  the  resultant  of  the 
applied  forces  and  two  forces  equal  to  half  this  force,  but 
opposite  in  direction,  to  the  extremity  of  the  _L  and  an 
equal  distance  on  the  other  side  of  the  centre  of  inertia 
respectively,  this  system  will  be  in  equilibrium.  But  this 
system  combined  with  the  resultant  of  the  applied  forces  is 
equivalent  to  a  single  force  acting  at  the  centre  of  inertia 
equal  and  parallel  to  the  resultant,  together  with  a  couple 
about  an  axis  through  the  centre  of  inertia.  Likewise,  revers- 
ing all  the  directions,  the  resultant  of  all  the  inertianal  forces 
is  equivalent  to  an  equal  and  parallel  force  acting  at  the 
centre  of  inertia,  together  with  a  couple  about  an  axis 
through  this  centre.  The  axis  of  the  couple  is  JL  to  the  plane 


MOTION    OF   RIGID    MASSES  25 

through  the  line  of  action  and  centre  of  inertia,  and  the 
intensity  of  the  couple  is  the  moment  of  the  resultant  of  all 
the  applied  forces  about  the  centre  of  inertia. 

If  the  resultant  of  the  applied  forces  passes  through  the 
centre  of  inertia  there  can  be  no  rotation.  A  homogeneous 
sphere  in  a  centrally  attracting  field  can  acquire  no  ro- 
tation and  is  said  to  be  centrobaric.  That  is,  the  resultant 
line  of  attraction  of  an  attracting  point  always  passes 
through  the  centre  of  inertia  of  the  sphere.  This  is  evident 
from  symmetry.  Likewise  no  body,  whatsoever  its  shape, 
can  acquire  a  rotation  in  a  uniform  parallel  field,  such  as 
the  field  at  the  earth's  surface. 

It  is  to  be  observed  that  generally  the  motion  of  the 
centre  of  inertia  is  not  the  same  as  if  the  whole  mass  were 
first  concentrated  into  its  centre  of  inertia  and  then  acted 
upon  by  the  field.  What  we  have  proved  is  that  for  any 
field  it  is  the  same  as  if  the  applied  forces  were  applied 
to  the  total  mass  at  the  centre  of  inertia,  parallel  to  their 
original  directions.  However,  in  certain  fields  the  result 
will  be  the  same  in  either  case.  In  uniform  parallel  fields 
such  will  obviously  be  the  case,  and  also  {v.  Art.  24)  when 
the  forces  tend  to  a  fixed  centre  and  vary  as  the  distance 
from  that  centre. 

If  we  define  the  centre  of  gravity  as  a  point  in  a  body 
such  that  when  fixed  the  body  is  not  rotated  by  the  field 
in  any  position,  it  is  evident  that  when  there  is  such  a  point 
it  is  the  centre  of  inertia,  but  that  generally  there  is  no  such 
point.  The  earth  being  a  spheroid  is  not  centrobaric  for 
central  fields  and  therefore  has  no  such  point.  The  sun's 
field  and  the  moon's  field  both  produce  rotations  of  the 
earth  which  result  in  the  precession  of  the  equinoxes. 
Usually  the  term  Centre  of  Gravity  is  taken  as  synony- 
mous with  Centre  of  Inertia.  Since  every  mathematical 
conception  should  have  a  single  name  and  as  there  are 
other  fields  than  gravitational,  it  would  seem  advisable 
to  employ  the  term  Centre  of  Inertia  alone. 

Using  polar  co-ordinates,  x  =  r  cos  i^,  y  =  r  sin  t?,  where 


26  NATURAL   PHILOSOPHY 


t?  is  the  angle  between  a  radius  in  the  %,  y  plane  and  the 
X  axis, 


sint?z?2  J.  Hence  the  couple  about  the  z  axis  is  llmr^d-  = 

^Zmr^.  The  integral  Swr2  is  called  the  moment  of  inertia 
of  a  body  about  an  axis  ±  to  r.  Letting  Swr2  =  Mr2, 
r  is  called  the  Radius  of  Gyration,  and  it  is  the  average 
radius  which  would  give  the  same  moment  of  inertia  if 
the  whole  mass  were  concentrated  at  its  extremity. 
A  couple  is  therefore  measured  by  the  moment  of  inertia 
into  the  angular  acceleration  about  an  axis. 

9.     Moments  of  Inertia 

Taking  some  point  in  a  body  as  origin  of  rectangular 
co-ordinates,  let  us  draw  radii  in  all  directions  from  the 
origin  of  such  lengths  that  the  moment  of  inertia  about 
any  radius  as  an  axis  shall  be  equal  to  the  square  of  the 

reciprocal  of  the  radius,  or  J  =   — ,  where  I  is  the  moment  of 

r2 

inertia.  The  locus  of  the  extremities  of  these  radii  will  be  a 

surface.   Designating  the  moments  of  inertia  about  the 

j_ 

=  i^,/,=2m(:^2  +  ^2)=  1 

7-22  '  7-3. 

is  2  Smr2,  a  constant,  where  r  is  the  distance  of  any  element 
from  the  origin,  and  ri,  r2,  rs  refer  to  the  momental 
surface.  We  have  taken  any  axes,  so  that  our  surface  has 
the  property  that  the  sum  of  the  squares  of  the  reciprocals 
of  any  three  X  radii  is  constant.  Such  a  property  belongs  to 
an  elHpsoid  alone.  For  let  ai,  jSi,  71,  :  ^2,  ^2,  72;  "3,  i^a,  73 
be  the  direction  angles  of  any  three  mutually  J_  radii 
referred  to  the  principal  axes  of  an  ellipsoid,  a,  b,  c.  Then 

1  COS2q;i  COS2j8i  COS27i 

r^2  =      a2     +     62      +  ""^T"  etc., 


axes  as  7*,  ly,  I^,  I^  =  Sw  (y^  +22)  =  :r^yly  =  2m  {x^  +  z^) 
,   Iz  =  2w  {x^  -\-  y2)  =,  ^,    The  sum  of  these  moments 


IMPULSIVE  FORCES  27 

Hence,  at  any  point  of  any  body  it  is  possible  to  con- 
struct an  ellipsoid  with  this  point  as  a  centre  such  that 
the  square  of  the  reciprocal  of  any  radius  is  equal  to  the 
moment  of  inertia  of  the  body  about  that  radius  as  an  axis. 
The  ellipsoid  corresponding  to  any  point  is  called  the  Mo- 
mental  Ellipsoid  for  that  point.  The  principal  axes  of  this 
ellipsoid  are  called  the  principal  axes  of  inertia  of  the  body 
for  that  point.  The  principal  axes  of  inertia  corresponding 
to  the  centre  of  inertia  are  called  simply  the  Principal  Axes 
of  the  body,  and  the  moments  about  these  axes  are  the 
Principal  Moments  of  Inertia.  Generally  the  three  principal 
moments  of  inertia  have  different  values  and  such  bodies 
are  said  to  be  triaxial.  When  two  of  the  moments  are  equal, 
the  body  is  biaxial  and  when  all  three  are  equal  the  body  is 
uniaxial. 

Taking  any  axis  about  which  we  wish  to  find  the  mo- 
ment of  inertia  as  the  z  axis  and  x,  y  as  the  co-ordinates  of 
the  centre  of  inertia  and  x^,  y^  as  the  co-ordinates  of  an 
element  referred  to  parallel  axes  through  the  centre  of 
inertia,  since  x  =  x-\-  x^  and  y  =  y  ^y^,  the  moment  of 
inertia  about  our  axis  is  Sm  (^2  _|-  ^2)  =  2m  (^2  -|.  yi) 
+  Sm  {x'"^  -h  ^'2)  since  ^mx^  =  llmy^  =  0.  Hence  the 
moment  of  inertia  about  any  axis  is  equal  to  the  moment 
about  a  parallel  axis  through  the  centre  of  inertia  plus 
the  moment  of  the  whole  mass  concentrated  into  the 
centre  of  inertia  about  our  axis.  The  moment  of  inertia 
about  an  axis  through  the  centre  of  inertia  is  therefore 
less  than  that  about  any  other  parallel  axis. 

10.     Impulsive  Forces 

We  have  already  seen  that  a  force  acting  continuously  is 
measured  by  the  mass  it  acts  upon  into  the  acceleration  it 
produces  in  the  mass  in  unit  time.  We  now  wish  to  deter- 
mine how  a  force  acting  only  for  a  brief  interval  may  be 
measured.  A  force  cannot  of  course  act  instantaneously  or 


28  NATURAL   PHILOSOPHY 

for  absolutely  no  time,  for  in  such  a  case  to  produce 
any  finite  effect  the  force  would  have  to  be  infinite  and 
there  is  no  such  thing  as  an  infinite  force. 

Two  elastic  balls  of  mass,  wi  and  W2,  and  velocities 
Vi  and  V2,  meet,  going  either  in  the  same  or  opposite 
directions.  We  shall  suppose  that  no  energy  is  lost  by  the 
impact  or  no  heat  developed.  The  velocity  of  the  first  ball 
is  changed,  not  instantaneously,  but  in  an  exceedingly 
short  interval  of  time,  from  vi  to  V]_\  and  that  of  the  second 
ball  from  V2  to  V2'.  During  the  short  interval  that  they  are 
in  contact  they  must  move  with  the  same  velocity,  v, 
and  this  velocity  is  the  average  velocity  while  the  change 
in  the  velocities  is  being  effected.  During  the  time  of 
contact  the  first  ball  has  changed  its  velocity  from  vi  to  ^1' 
and  its  average  velocity  during  this  time  must  have  been 


2 


while  the  average  velocity  of  the  second  ball  was 


1)2   -\-  V2' 

— X .    The  kinetic  energy  of  the  system  remains  un- 

1           J         XI    ^  m\vi2        m2V2^        mxvi'^        m2V2^ 
changed,  so  that  -^ 1 ^^   =       2 ' 2      * 

Hence  mi  r  ^  2  ^  )  ^^^  ~  "^^'^  "^^  \~Y^  )  ^^^'  ~  '^^^' 
But  — ^ —  =  ^ —  =  V,  and  wi(^i  —  Vi)  =  m2\V2  —  V2). 

We  have  thus  the  measure  of  a  force  which  acts  for  a 
short  time  and  which  is  called  an  impulsive  force  or  an 
impact.  The  measure  is  the  increase  (or  decrease)  of  the 
momentum  which  it  produces  in  a  body. 

11.     Pendulum 

We  have  seen  that  the  time  of  a  complete  small  oscilla- 
tion of  a  suspended  particle  is  lir-J—.     If  we  have  a  rigid 

body  of  mass  M  oscillating  about  a  horizontal  axis,  the 
gravitational  couple  is  Mgh  sin  t?,  where  h  is  the  distance 


PENDULUM  29 

of  the  centre  of  inertia  from  the  axis  and  t?  the  angle  it 
makes  with  the  vertical.  If  I  is  the  moment  of  inertia 
about  the  axis,  !-&  =  —Mgh  sin  ^  (1).  li  k  is  the  radius  of 
gyration  about  a  parallel  axis  through  the  centre  of  inertia, 
I  =  M{k2  -\-  h^).  For  a  small  oscillation  sin  ^  is  sensibly- 
equal  to  z?.  Integrating  (1),  I?  =  -J — ^ —  V??i2  _  ^2, 
where  t9i  is  the  maximum  excursion.  Integrating  again, 
t    =  ^^IjhJjL  f  y  —  sin      Y  I  •     The  time  of  a  complete 

oscillation   is    T  =  lir  -J      "^       and   the   length   of   the 

T      gh 

J^2   J-  /j2 

equivalent  simple  pendulum  is  I  =  7 

If  we  wish  to  find  the  time  for  any  amplitude  we  may 
proceed  as  follows:  Integrating  (1)  ??  = 

Jl  V2(cost?-cos^o)  =   iJ^Jsin^^-  sin2!L. 
Let  sin  ^  sin  <^  =  sin  y »  where   (p  is  an  auxiliary  angle. 
When  t?  =  0,  ^  =  0  and  when  ^  =  1^0,  <P  =  w-'    Hence  dt  = 


€ 


dip 


^l  —  sin2-^sin2<p 


2 

By  the  binomial  theorem  the  radical  can  be  developed  into 
1  ^  13?? 

1  +  -^  sin2  -^  sin2^  +  "2'    4"  ^^^*  Y  ^^^^  ^  "^  ^^^' 

TT 

Now  J'   sin2»  ^rf^  =  l^i.  -|.  A  .  .  .2!L^y 

Hence  the  time  of  a  complete  swing  is 

r...Vf['4)"-^Gi)"''4'  +  -] 

From  this  rapidly  converging  series  we  can  determine  the 
time  for  any  amplitude  as  closely  as  we  please. 


30 


NATURAL   PHILOSOPHY 


12.   Centres  of  Oscillation  and  Percussion 

An  impulsive  force  acting  upon  a  body  obviously  im- 
parts to  it  an  impulsive  velocity  of  the  centre  of  inertia,  or 
an  impulsive  momentum  of  the  entire  mass  considered  as 
concentrated  at  this  centre,  together  with  an  impulsive 
moment  of  momentum  about  an  axis  through  this  centre. 
If  a  body  (Fig.  6)  is  struck  a  blow  at  B  in  the  direction 
AB,  the  translational  motion  of  the  centre  of  inertia,  G,  is 
the  same  as  if   the   entire   mass  were 
concentrated  there  and  the  force  acted 
on  it  parallel  to  its  direction,  and  the 
rotational  moment  of  momentum  about 
an  axis  through  G  is  the  same  as  if  this 
centre  were  fixed.  The  impulsive  veloc- 
ity, V,  of  the  centre  of  inertia,  parallel  to 
AB,  will  move  it  a  distance  vdt  in  the 
first  small  interval  of  time,  and  the  ro- 
tation about  an  axis  through  G,    JL  to 
the    plane   ABG,  will    turn    the    body 
through  an  angle  d-^  in  this  time.  This  is 
equivalent  to  a  rotation  about  a  parallel  axis  through  some 
point  0  in  the  line  06",  ±  to  AB.  Calling  OP,  /,  and  OG,  h, 
the  moment  of  the  blow  F  about  G  is  measured  by  the 
impulsive  moment  of  momentum  about  the  axis  through 

(7,  or  F{l-h)  =  Mk2  t?  and  ^  =    ^^^  "  ^^ 


Mk^ 


-,  where  k  is  the 


radius  of  gyration  about  the  axis  through  G.  The  blow  F 
is   also   measured  by  the  impulsive   momentum  of  the 


centre  of  inertia,  or  F 
k2  +  h2 


Mv  =  Mh&  =  Mh 


F{1  -  h) 
Mk2 


,  and 


/  = 


h 


The  axis  of  spontaneous  rotation  for  the 


first  instant,  or  the  instantaneous  axis,  is  therefore  at  a 
distance  from  the  line  of  the  blow  equal  to  the  length  of  the 
equivalent  simple  pendulum.  Relative  to  an  axis  through 
0,  the  point  P  is  the  centre  of  oscillation,  and  it  is  also  a 


HARMONIC   MOTION  31 

centre  of  percussion,  for  a  blow  applied  at  this  distance 
from  the  axis  of  spontaneous  rotation  would  cause  no 
reaction  upon  it.  If  we  strike  a  ball  with  a  bat  or  a  tree 
with  an  axe  at  a  distance  from  our  hands  equal  to  the 
length  of  the  equivalent  simple  pendulum,  we  experience 
no  "sting";  otherwise  we  do.  The  subsequent  motion  of 
the  body  consists  simply  of  the  initial  impulsive  velocity 
of  the  centre  of  inertia,  and  the  impulsive  rotation  about 
the  axis  through  it,  with  the  superposed  effects  of  sub- 
sequent forces.  The  trajectory  of  a  struck  ball  is  determined 
by  the  initial  velocity  of  the  centre  of  inertia,  the  spin 
not  being  influenced  by  gravity. 

li  GP  =  h\  I  =  h  +  h^  and  hh^  =  k^.  Hence  the  points 
0  and  P  are  mutually  centres  of  oscillation  and  percussion, 
one  for  the  other,  and  the  times  of  oscillation  (under 
gravity)  are  the  same  for  either  suspension.  This  gives  a 
means  of  determining  the  length  of  the  equivalent  simple 
pendulum. 

13.   Harmonic  Motion 

Distant  bodies  influence  each  other  by  gravitational 
(longitudinal)  waves  through  the  ether.  Since  such  forces 
radiate  equally  in  all  directions  from  a  central  point,  the 
intensity  of  the  force  must  fall  off  as  the  square  of  the 
distance,  or  the  simple  geometrical  law  must  hold.  Since 
two  equal  bodies  radiate  twice  the  energy  of  a  single  body, 
the  force  must  be  proportional  to  the  mass,  for  the  same 
distance.  Again  the  action  of  such  a  force  upon  two  equal 
distant  bodies  must  be  twice  the  action  upon  a  single  body. 
Hence  we  have  almost  axiomatically  Newton's  law  of 
attraction — "All  bodies  attract  each  other  with  a  force 
proportional  to  the  product  of  their  masses  and  inversely 

as  the  square  of  the  distance,  or  /  =  — r—  •        The  action 

is  purely  mutual,  and  the  forces  urging  each  body  are  pre- 
cisely equal,  and  opposite. 

Let  us  consider  the  mutual  force  between  a  homogeneous 


32  NATURAL   PHILOSOPHY 

Spherical  shell  and  unit  mass  at  any  point.  Let  a  be  the 

density  of  the  matter  on  the  shell,  or  mass  per  unit  surface, 

and  h  the  distance  of  the  point  from  the  centre  of  the 

shell,  r  its  radius,  p  the  distance  of  the  point  from  any 

element  of  the  shell  and  ^  the  angle  between  any  radius 

and  h.  From  symmetry  it  is  evident  that  the  resultant 

attraction  of  the  shell  must  be  along  h.  An  elemental  ring, 

concentric  with  the  axis  h,  has   a   mass    l-Kar^   sin   §d^. 

The  component  along  k  of  every  such  elemental  ring  is 

27r(Tr2  sin  ??  ,,  ..    ,. 
{h  —  r  cos  z?)  d^. 


pdp 

rh 


Since       p2  =  h^  -{-  r^  —  2rh  cos  ??,  sin  ^  d^  = 

and  h  —  r  cos  t?  =  x; 

2h 

The  total  attraction  is 

TTcrr         fft  +  r(/,  +  ^)  (k  -  r)    ,      ,    ircrr         r*  +  ^  ,     ,.. 

■^.  -  J    ? —  '^  ^  ^.  -  J    '^^^^'^^;^ 

There  are  three  cases:     li  h  >  r,  we  have  /  =  — t-—: 

\i  h  =  r,  f  =  4x0-:  and  if  h  <  r,  we  have  to  take  the 
limits  as  r  -\-  h  and  r  —  h,  and  the  integral  becomes 

Hence  any  homogeneous  spherical  shell  attracts,  and  is 
attracted  by,  any  external  mass  precisely  as  if  its  mass 
were  concentrated  at  its  centre,  but  within  the  shell  it 
exercises  no  attraction,  or  there  is  no  force.  And  any 
sphere  made  up  of  homogeneous  spherical  shells  attracts  in 
the  same  way. 

Let  us  suppose  a  homogeneous  sphere  with  two  diame- 
ters bored  out   _L  to  each  other.  The  attraction  of  the 

sphere  on  unit  mass  at  the  surface  is  -^r  =  -^ — ,  and  the 

attraction  at  any  level  in  the  interior  is  ^  ,  where  r  is 
the  distance  from  the  centre.  Hence  if  we  drop  unit  mass 


HARMONIC   MOTION  33 

into  one  of  these  holes,  it  will  oscillate  harmonically  be- 
tween two  diametrically  opposite  points  of  the  surface,  the 
motion  being  harmonic  because  the  force  is  proportional 


to  the  distance  from  the  centre.  Let  2^-^  hek.  If  we  pro- 
ject the  mass  along  the  surface  with  a  velocity,  v  =  kR, 
which  makes  the  centrifugal  force  just  equal  to  the 
attraction,  it  will  revolve  about  the  sphere,  just  grazing 
the  surface.  The  time  for  the  outside  mass  to  traverse  a 

quadrant  is  -^  ,and  the  time  to  reach  the  centre  is  the  same. 


1 


k2r,  r  =  k^  R2  —  r^  and  ^  =  rsin     -^ 


R 


Hence  the  outside  and  inside  masses  will  regularly  meet  as 
the  outside  mass  passes  over  each  hole.  Using  the  holes  as 
axes  of  X  and  y,  the  positions  of  the  inside  masses  will  be 
the  co-ordinates  of  the  outside  mass. 

The  inside  bodies  execute  simple  harmonic  motions, 
while  the  outside  body  executes  a  compound  harmonic 
motion  made  up  of  two  equal  simple  harmonic  motions  ± 


1        -If 
to  each  other.  Since  ^  =  ^-  sm    — 

k  To 


where  Yq  is  some 


level  within  the  sphere  from  which  we  drop  a  body,  the  body 

thus  dropped  will  have  the  same  period  as  if  dropped  from 

the  surface.   Thus  the  period  of  any  harmonic  motion 

is  independent  of  the  amplitude  of  the  motion.  It  is  for  this 

reason  that  such  a  motion  is  called  harmonic.  The  vibrating 

parts  of  all  musical  instruments  execute  harmonic  motions. 

Otherwise    the    period    (pitch)  would    change    with    the 

intensity  (amplitude)  and  music  or  harmony  would  become 

impossible. 

Two  ±  simple  harmonic  motions  with  the  same  period 

and  amplitude,  and  one  of  the  motions  a  quadrant  in 

advance   (or  behind)   of  the  other,  result  in  a  circular 

harmonic  motion.  If  the  amplitudes  are  not  equal  we  have 

TT  1      .    ~^^ 

elliptic  harmonic  motion.  Since  ^  =  -^  +  -r-  sin     ^^  for 


34 


NATURAL   PHILOSOPHY 


one  mass  when  ^  =    r  sin     -  for  the  other,  we  may  write 


y  =  ro  sin  kt  and 


R  sin 

X2 


("  -  r) 


R  cos  kt. 


R2 


+  21  = 


1,  or  the  path  for  the  com- 


pound motion  is  an  ellipse.  The  angle  in  these  expressions 
is  called  the  phase  of  the  motion  and  the  difference  be- 
tween the  phases  of  the  two  components  is  always  y- 

The  reciprocal  of  the  period  is  called  the  frequency,  or  the 

number  of  vibrations  in  a  second.  The  complete  period  is 

27r 

— .  It  is  evident  that  any  number  of  simple  harmonic 

motions  making  any  angles  with  each  other,  but  all  having 
a  common  centre,  can  be  compounded.  If  there  is  a  com- 
mon period  the  result  will  be  a  steady  elliptic  motion. 
If  the  periods  are  different  the  resultant  motion  will  be 
continually  shifting,  forming  what  are  known  as  Lissajon's 
curves.  If  the  periods  have  a  common  multiple  the  changes 
will  periodically  repeat  themselves. 

14.     Tidal  Forces 

A  homogeneous  ring  revolves  about   an  attractional 
centre,  S,  which  is  in  its  plane,  and  the  plane  of  the  ring 

is  ±  to  its  orbit.  The 
distance  SC  =  D  is 
constant  and  (p  is  the 
angle  any  element  of 
the  ring  makes  with 
CA .  D  is  so  great  that 
all  lines  from  5  to  the 
ring  may  be  regarded 
as  sensibly  parallel. 
The  centrifugal  force 
for  any  element  is  (D  +  r  cos  cp)  yj/^rdip,  where  yp  is  the 
orbital  angular  velocity.   Integrating,   we  find  that  the 


Fig.  7. 


TIDAL   FORCES  35 

centrifugal  force  for  the  outer  half  is  TrrDxp'^  +  2r2^2^  and 
for  the  inner  half  7rrD\l/^  —  2r2^2.  Calling  the  mass  of  the 
ring,  M,  these  two  centrifugal  forces  are 

-^  Z)i/'2  ^  —-.  —  p  and  -^  DrP^  "  T'  —  "l"^' 

Ir 

—  is  the  distance  of  the  centre  of  inertia  of  a  half  ring  from 

C,  so  that  the  rotational  centrifugal  force  for  each  half  of 
the  ring  is  the  same  as  if  its  mass  were  concentrated  at  its 
centre  of  inertia. 

Let  /  be  the  acceleration  due  to  the  central  attraction 
at  unit  distance.  Then  the  total  attraction  of  the  ring  is 

Jfrdip r2/  sin  (p _ 
(D  +  r  cos  <py'   ~   (D2  _  r2)  {D  -\- r  cos  ip) 

-r=- —      .^  cos      "F— -^ •     Taking  the  limits  for  the 

(£)2  —  r2)|  D  +  r  cos  ^  ^ 

outer  half,  and  considering  that  {D^  —  r^)  is  nearly  equal 

—  ^T  IT 

to  D2  and  the  angle  cos  j^  is  nearly  equal  to  -x,  we  have 
for  the  attraction  on  the  outer  half  —  -^  H — ^rr--  Like- 
wise  the  attraction  on  the  inner  half  is  —  r=^ ^-  If  the 

L/2  JJ^ 

total  centrifugal  force,  which  is  MDyj/^,  balances  the  total 

Mf 
attractional  force,  which  is  —  jr^ ,  all  the  forces  will  be  in 

equilibrium,  and  the  motion  will  be  equivalent  to  a  mass 
concentrated  at  C  revolving  about  5  at  a  constant  distance 
with  constant  angular  velocity.  But  we  have  a  force  acting 

on  the  outer  half, (  ^  +  i/'^  ),  which  is  balanced  by  an 

equal    and    opposite    force    acting    on    the   inner    half, 

(-^  H-i/'^V     These  two  forces,  therefore,  tend  to 

IT    \V'^  f 

pull  the  ring  apart  in  either  direction  from  the  centre,  the 
inner  half  being  urged  towards  the  attracting  body,  while 


36  NATURAL   PHILOSOPHY 

the  outer  half  is  urged  directly  away  from  it.  If  the  ring  is 

not  perfectly  rigid,  and  perfectly  rigid  bodies  do  not  exist, 

it  will  be  deformed  into  an  oval   pointing  towards  the 

Mr     f 
attracting   body.    The   differential    forces    jr-^    and 

^  are  called  Tidal  Forces.  As  a  planet  can  be  con- 

sidered  to  be  made  up  of  rings  which  are  concentric  on  a 
line  Jl  to  the  plane  of  our  ring,  it  is  evident  that  these 
tidal  forces  are  continually  deforming  it  into  an  oval 
which  always  points  towards  the  attracting  body.  The 
moon  and  the  sun  both  deform  the  earth  into  ovals  point- 
ing towards  them.  The  longest  axis  of  the  oval  is  towards 
the  attracting  body,  while  the  shortest  axis  is  ±  to  the 
orbit. 

The  effect  upon  the  rotation  is  similar  to  a  friction 
band  tending  to  slow  the  rotation  down  to  coincidence 
with  the  revolutional  period,  and  eventually  this  must 
occur.  The  rotation  cannot  fall  below  the  revolution  for  in 
that  case  the  tidal  band  would  accelerate  it  into  coinci- 
dence again.  The  effect  in  the  case  of  the  earth  is  very 
slight  but  the  height  of  this  solid  tide  may  possibly  approxi- 
mate a  foot.  We  do  not  know  exactly  what  it  is.  It  is 
constantly  at  work  and  it  has  very  important  and  far- 
reaching  effects  which  we  shall  discuss  later.  The  earth  is 
always  rotating  about  two  axes  —  one  being  the  diurnal  or 
polar  axis,  the  other  being  the  precessional  axis  which  is  ± 
to  the  former.  The  effect  of  the  tidal  brake-band  is  to 
oppose  these  two  rotations,  with  the  result  in  one  case  of 
lengthening  the  day  and  in  the  other  of  changing  the  inclina- 
tion of  the  earth's  axis.  The  oceans  constitute  a  thin 
skin  of  water  covering  about  ^  of  the  earth's  surface  and 
they  are  of  course  subject  to  the  tidal  action  we  have  just 
discussed.  Theoretically,  the  effect  of  these  ocean  tides  is 
to  decrease  the  earth's  motion,  but  compared  with  the 
solid  tides  of  the  earth's  mass  the  effect  is  insignificant. 
Thus  while  the  solid  tides  will  undoubtedly  bring  the 


ATTRACTIONAL   AND    MOLECULAR   RIGIDITY      37 

rotation  and  revolution  into  coincidence  in  some  finite 
time,  the  ocean  tides  could  only  effect  this  in  anin- 
finite  time.  As  our  earth  has  not,  and  will  not  exist 
for  an  infinite  time,  such  ocean  effects  are  wholly  negli- 
gible. 

When  two  bodies  are  near  each  other,  it  will  be  seen 

that  the  terms  containing  -^  and  xj/^  become  very  large, 

and  the  rending  force  may  reach  a  limit  beyond  which  the 
two  bodies  would  be  torn  apart.  Since  these  forces  are 
also  proportional  to  r,  this  limit  would  be  reached  sooner 
in  larger  bodies. 

The  particles  of  an  attracted  body,  instead  of  executing 
circles  about  the  rotation  axis,  are  forced  to  move  in 
ovals  and  as  their  distances  from  each  other  are  now 
greater  and  now  less,  they  are  continually  kneaded, 
stretched  and  compressed,  and  heat  is  produced,  just  as  a 
hammered  bar  becomes  hot.  All  this  heat  energy  which  is 
dissipated  is  produced  at  the  expense  of  the  rotation. 
When  the  rotational  and  revolutional  periods  coincide 
this  action  ceases.  Such  a  coincidence  of  periods  probably 
exists  in  all  the  satellites  of  our  system,  and  in  Venus  and 
Mercury.  Our  moon  is  a  near  example. 

15.    Attractional  and  Molecular  Rigidity 

In  a  very  large  mass,  such  as  the  earth,  there  is  no  in- 
herent elasticity  of  figure  such  as  exists  in  small  masses  and 
which  is  due  to  the  interaction  of  adjacent  molecules. 
A  small  mass  can  have  any  shape  while  a  large  mass,  of 
itself,  has  only  one  shape,  viz.,  a  sphere.  It  would  be  im- 
possible for  a  body  as  large  as  the  earth  or  the  moon  to 
have  the  shape  of  a  bar.  We  must  thus  distinguish  between 
two  kinds  of  elasticity,  or  force  tending  to  restore  a  body 
to  its  original  shape  after  deformation. 

Our  conceptions  of  matter  are  formed  from  the  con- 
ditions under  which  it  exists  at  the  earth's  surface,  but  it 
is  unsafe  to  project  these  ideas  into  places  where  the 


38  NATURAL   PHILOSOPHY 

conditions  differ  widely.  A  solid  spring  regains  its  shape 
because  of  the  cohesion  between  its  adjacent  molecules  and 
it  possesses  only  a  proximate  molecular  rigidity.  As  the 
size  increases  this  kind  of  rigidity  becomes  of  less  im- 
portance until  it  is  entirely  overcome  by  self -gravitation 
and  the  mass  flows  into  a  sphere.  A  fluid  is  etymologically 
that  which  flows,  so  that  in  saying  that  the  mass  flows  into 
a  sphere  we  imply  that  it  is  no  longer  a  solid  body  but 
a  fluid  body,  although  at  the  surface  small  portions  may 
still  be  regarded  as  solid.  When  we  melt  a  solid  it  loses 
largely  its  molecular  rigidity  and  assumes  a  form  of  equi- 
librium depending  chiefly  upon  gravitational  forces.  The 
matter  in  the  interior  of  a  large  body,  such  as  the  earth,  is 
thus  certainly  fluid  or  plastic  because  the  shape  it  assumes 
is  due  solely  to  the  enormous  gravitational  pressures  and 
not  at  all  to  any  molecular  rigidity.  The  question  as  to 
whether  the  interior  of  the  earth  is  solid  or  liquid  is 
founded  upon  a  confusion  between  general  attractional 
rigidity  and  molecular  rigidity.  It  is  certainly  plastic  and 
very  hot  —  probably  well  above  4000°C  from  the  centre 
to  a  few  miles  from  the  surface.  For  all  practical  pur- 
poses, it  is  therefore  in  a  molten  condition. 

A  very  large  body  must  necessarily  have  a  very  high 
rigidity,  exceeding  that  of  a  very  small  body,  but  the 
resistance  to  deformation  is  of  a  very  different  nature  in 
the  two  cases  and  it  is  difficult  to  compare  them.  The 
earth  has  a  high  rigidity  but  it  is  manifestly  improper  to 
say  that  it  has  the  rigidity  of  a  globe  of  steel  or  of  a  globe 
of  glass  of  equal  size. 

We  may  determine  the  rigidity  of  a  glass  marble,  which 
is  the  rigidity  of  its  proximate  molecules,  but  if  we  in- 
creased it  to  the  size  of  the  earth,  this  particular  kind  of 
rigidity  would  become  insignificant  in  comparison  with  its 
general  attractional  rigidity.  A  small  piece  of  glass  might 
have  any  shape,  but  a  large  piece  could  have  only  one 
shape.  A  further  difficulty  occurs  in  that  a  globe  the  size 
of  the  earth  could  not  exist  as  glass  or  steel  throughout. 


DENSITY   OF   THE   EARTH  39 

The  enormous  condensation  towards  the  centre  would 
necessarily  result  in  changing  the  kind  of  matter.  We  have 
previously  referred  to  a  suggestion  that  the  ether  may  be 
the  ground  stuff  out  of  which  all  the  "elements"  have  been 
formed  under  varying  conditions  of  pressure  and  tempera- 
ture. This  is  merely  a  surmise,  though  perhaps  a  natural 
one  when  we  see  denser  elements  spontaneously  splitting 
into  lighter  ones.  We  know  that  the  average  density  of 
the  earth  is  5^  times  that  of  water,  while  the  surface 
density  is  only  about  2^.  There  must  be  much  matter  in 
the  interior  which  is  much  denser  than  the  surface  matter 
and  it  seems  probable  that  such  matter  may  be  like  some 
of  the  denser  elements  with  which  we  are  familiar.  A 
considerable  part  of  the  earth's  interior  is  probably  iron. 
Magnetic  phenomena  point  to  much  paramagnetic  matter 
■ —  iron,  nickel,  cobalt.  Siderites  seem  to  be  fragments  of 
some  former  body,  and  their  usual  iron-nickel  composition 
suggests  that  many  bodies  in  the  universe  are  automatic- 
ally largely  composed  of  these  substances. 

16.     Density  of  the  Earth 

The  matter  of  the  earth  was  probably  once  homogeneous, 
but  by  self -gravitation  has  condensed  to  its  present  state. 
The  density  of  this  matter  must,  on  the  whole,  increase 
regularly  and  progressively  from  the  surface  to  the  centre. 
What  the  exact  law  is,  we  do  not  know.  Laplace  assumed, 
as  a  hypothesis,  that  the  increase  of  the  square  of  the 
density  was  proportional  to  the  increase  of  the  pressure. 
That  is  to  say  the  compression  gradually  decreases  as  the 
density  increases.  While  perhaps  not  the  actual  law,  it 
would  seem  that  Laplace's  formula  must  represent  the 
actual  conditions  to  a  close  approximation,  even  bearing 
in  mind  the  possibility  that  the  condensation  may  at 
certain  stages  have  been  per  saltum,  as  where  an  element 
may  have  changed  suddenly  into  a  denser  one. 

Let  r  be  the  distance  of  any  level  from  the  centre,  p 
and  /  the  density  and  acceleration  at  that  level,  and  pa  the 


40  NATURAL   PHILOSOPHY 

average  density  of  all  the  matter  within  that  level.  Taking 
the  direction  outward  as  positive, 


■^  ""  ~  r2    I    ^'^f^^^^^  =  ~  -T  ^Pa- 


3 


Hence  rDrPa  =  3  {p  —  Pa)  (2). 

Laplace's  formula  is  pdp  =  kdp,  where  p  is  the  pressure 
and  k  is  some  constant,  dp  =  pf.dr.  Whence  DrP  =  kf  (3). 
p  is  a  function  of  r,  or  for  any  value  of  r  there  is  a  single 
definite  value  for  p. 

p  =  ^  (r).  By  Maclaurin's  theorem 

P  =  <P  (o)  -}-  <pi  (o)  r  +  <P^^  (o)  2j  +  etc. 


DrP  =  <p^{r)  =  kf.     <p^^  (r)  =  kDrf  =  ^rk 


(j'^-') 


2  (P   -  Pa)    _^- 


^iii(r)  =  ^Trk  \j  DrPa  -  kf\  =47r^r 

,Kr)=-[Mfl(l)+,uw+i^]. 

(p{o)  =  pc,  where  pc  is  the  density  at  the  centre. 
<p^{o)  =  0,  since  the  force  at  the  centre  is  zero. 

4:Trk 

<p^^  (o)  = :T-pc-  v^^K^)  becomes  indeterminate  when 

r  is  zero,  but  by  differentiating  numerator  and  denomi- 
nator of  the  indeterminate  term,  we  obtain  zero  as  the 
limiting  value. 

Limit  (piv{r)  =  —  Sirkip^^ir)  —  4c(piv(r), 
{r  =  6>)  (r  =  o)       {r  =  o) 

35  7r2yfe2 
and  limit  <piv{r)  =  — t-e —  Pc-    Similarly  we  find  that  ^»(o) 

(r  =  0) 
—  0.  All  the  odd  orders  of  <p{o)  are  zero  and  all  the  even 
orders  are  multiples   of  pc  with  increasing  powers  of  k 
in  the  coefficient.  Thus  (pvi{o)  is  a  multiple  of  —  ^^pc,et  c. 
^  is  a  very  small  quantity  so  that  we  can  neglect  higher 


DENSITY   OF   THE   EARTH 


powers  than  the  first.  Hence  we  have 
Denoting  the  average  density  of  the  body  by  pa, 


41 


(4) 


and 


k  = 


(-» 


(5). 


and 


IttRP- 

Hence,  5pa  =  2pc  +  Sp^  (7),  where  Ps  is  the  surface  density. 
We  have  then  the  general  law  that  for  any  large  body 
(sphere),  five  times  the  average  density  is  equal  to  twice 
the  central  density  plus  three  times  the  surface  density. 
In  the  case  of  the  earth,  many  surface  rocks  have  a  density 
around  2.75.  Assuming  this  as  the  average  surface  density, 
which  is  about  half  the  average  density  of  the  whole  earth, 


we  have  Pc  ==  y  ^^' 


or  the  density  at  the  centre  of  the  earth 


is  3>^  times  the  surface  density,  or  it  is  9.63  times  the 
density  of  water. 

From  Equa.  (1)  it  will  be  seen  that  if  the  surface 
density  of  any  body  is  less  than  ^  of  its  whole  average 
density,  there  will 
be  a  level  between 
the  surface  and  the 
centre  where  fzpa^ 
p,  and  DrJ  =  0.  At 
this  point  /  is  a 
maximum  and  the 
p  curve  has  a  point 
of  inflexion.  At  this 
particular  level  the 

body  exerts  a  greater  attraction  than  at  any  other  distance 
from  the  centre,  and  we  shall  call  it  the  critical  level.  In 
the  earth  diagram,  Fig.  8,  this  level  is  shown  at  A . 


42  NATURAL   PHILOSOPHY 


Let  X  =  AS  =  R  —  r,  where  r  =  CA. 

5    r2 
R   (Pa  -  2ps) 


2      {Pa   -      Ps) 


and 


Pa    =  Pc 


3   R2        ^ 


Combin- 


5_  r2_  _2_ 
3  R2       3 

ing  these  equations  we  find  that  r  =  .94i?.  Or  the  critical 
level  of  the  earth  lies  about  240  miles  below  the  surface. 
The  attraction  therefore  increases  gradually  up  to  this 
point  after  which  it  decreases  regularly  to  zero  at  the 
centre.  Only  bodies  of  a  certain  size  can  have  a  critical 
level  and  the  fact  that  this  level  is  so  near  the  earth's 
surface  would  seem  to  indicate  that  bodies  of  a  little  less 
size  would  have  none.  The  moon  probably  has  no  critical 
level,  or  its  attraction  is  a  maximum  at  its  surface, 
decreasing  both  towards  and  away  from  the  centre. 

Taking  as  our  units,  feet  and  seconds,  the  unit  of  mass 
will  be  that  mass  which  attracts  a  like  mass  at  unit  dis- 
tance with  unit  force,  or  a  force  giving  it  unit  acceleration 
in  unit  time.  The  earth  attracts  unit  mass  at  its  surface 
with  a  force  of  32  units.  It  would  therefore  attract  unit 
mass  at  unit  distance  with  32i^2  units  of  force,  and  unit 

M 
mass  IS  -ry^,  where  M  is  the  mass  of  the  earth.  The  mass 

47r 
of  the  earth  is  equivalent  -r- ^^  X  ^-^  c^-  ^^-  of  water.  Hence 

unit  mass  is  equivalent  to  .72R  cu.  ft.  of  water,  where  R  is 

expressed  in  feet.  Or  the  matter  of  some  14  million  cu.  ft. 

of  water  concentrated  into  a  point  would  attract  a  like 

mass  at  the  distance  of  a  foot  with  a  force  which  would  give 

it  a  velocity  of  a  foot  a  second  for  every  second  that  it  acted. 

Unit  density  is  this  unit  mass  concentrated  into  a  cubic 

foot.  Hence,  expressed  in  these  units,  the  average  density 

7  68 
of  the  earth  is-^  =  3.68  x  10"^,  and  the  surface  density 
K 

•  3.84    ^    ^,  ^    .   /      , 

IS  — ^  .  In  the  same  way  we  find  that  the  pressure  at  the  cen- 
tre of  the  earth  is  about  49  million  pounds  per  square  inch. 


GREEN'S   THEOREM  43 

We  shall  refer  here  briefly  to  certain  arguments  for  the 
solidity  of  the  earth's  interior.  With  the  feeble  pressures 
available  in  our  laboratories  it  has  been  found  that  bodies 
which  expand  on  melting  have  their  melting  points  raised 
by  pressure  while  bodies  which  contract  on  melting  have 
their  melting  points  lowered  by  pressure.  It  is  argued  that 
the  enormous  pressures  in  the  interior  will  prevent  the 
matter  from  liquefying.  Further,  certain  surface  rocks 
have  been  melted  and  it  has  been  found  experimentally 
that  there  is  a  slight  expansion.  On  the  other  hand  the 
molten  lava  lake  of  Kilauea  is  often  skimmed  over  with  a 
solid  crust  just  as  a  lake  of  water  is  covered  with  a  sheet 
of  ice  in  winter,  and  in  both  cases  the  crusts  are  readily 
supported  by  the  underlying  liquid.  This  would  indicate 
that  the  deeper  matter,  from  which  the  lava  comes,  con- 
tracts on  melting.  Again  iron  contracts  on  melting  and 
there  is  little  doubt  that  iron  forms  a  considerable  part  of 
the  earth's  interior.  According  to  the  argument,  therefore, 
a  very  considerable  part  of  the  interior  matter  may  have 
its  melting  point  lowered  by  the  enormous  pressures  and 
is  therefore  liquid.  But  such  arguments  are  entirely 
inapplicable.  Beyond  a  certain  limit  of  pressure  all  matter 
becomes  fluid,  or  flowing,  and  plastic.  The  interior  of  the 
earth  is  certainly  plastic  and  very  hot,  and  is  therefore 
to  all  intents  and  purposes  in  a  molten  condition.  The 
extrusion  of  molten  rock  from  all  parts  of  the  earth's 
surface  points  strongly  to  such  a  general  condition.  Equally 
futile  is  the  argument  that  the  earth  must  be  solid  to  have 
its  high  rigidity.  The  earth  has  a  very  high  gravitational 
rigidity,  but  no  molecular  rigidity  except  at  the  surface. 
Surface  conditions  of  matter  cannot  be  projected  into 
the  interior. 

17.     Green's  Theorem 

Let  us  suppose  that  within  a  closed  surface,  5,  1^  is  a 
function  which  has  a  single  finite  value  for  every  point  of 
our   enclosed   space,    and   varies   continuously    (without 


44  NATURAL   PHILOSOPHY 

abrupt  change)  in  any  direction.  Choosing  any  axes,  a 
line  parallel  to  the  x  axis  must  cut  such  a  surface  an  even 
number  of  times. 


s 


.  dx  =  W2  —  Wi,  where  W\  is  the  value  of  W  at  the 

point  of  entrance  and  W2  its  value  at  the  point  of  exit. 

I    I    I  ~a~ '  ^^^y^^  ^  I    I  Wdydz,  the  volume  integral  on 

the  left  being  taken  throughout  the  whole  closed  space,  and 
the  surface  integral  over  the  whole  surface.  Representing 
the  volume  element,  dxdydz,  by  dr  and  the  surface  element 
dydz  by  dS  cos  (nx) ,  where  dS  is  the  element  of  the  closed 
surface  cut  out  by  an  elementary  parallelopiped  and 
(nx)  is  the  angle  between  x  and  the  normal  to  the  surface, 

drawn  inward,  j    I    1  -^  dr  =  1    j  W  co^  {nx)  dS .  W  is  2iny 

continuous  finite  point  function,  and  if  U  -r-  be  such  a 

dx 

function,  we  can  substitute  it  for  I^,  or  j    I    j  — (  U  —  \dT 

=  I  I  L^  -T—  cos  {nx)  dS  (1),  with  similar  expressions  for 
y  and  z.  Adding, 

(fcm^  +  ^^  +  ^^y. 

J  J  J    \  dx     dx  dy     dy  dz      dz  J 

""III  "^ —  ^^^  ^^^^  "^  ~^  ^^^  ^^^^  "^  ~^  ^^^  ^'^^^  ) 

This  result  is  known  as  Green's  Theorem.  The  derivatives 
here  are  partial  as  indicated  by  the  notation  and  obviously 

— —  cos  (nx)  +  ^--  cos  {ny)  +  -r—  cos  {nz)   =  -7 — ,  where 
dx  dy  dz  dn 

dV 

-r —  signifies  differentiation  in  the  direction  of  the  normal. 
dn 


GAUSS'   THEOREM  45 


Hence  we  can  write  (2) 


dU    dV         dU    dV         dU    dV 
dx     dx  dy     dy  dz      dz 


Since,  by  symmetry,  U  and  V  are  interchangeable,  we  have 

r  r  c[vf—-\-—4-—\-  u{—4-—-^-—\ 

J  J  J  L  v^^  ^y^     ^^v      v^^  ^y^     ^^v 

(3).    Equa.  (3)  is  known  as  Green's  Theorem  in  its  second, 
form. 

18.     Gauss*  Theorem 

Since  every  atom  is  always  in  motion  it  radiates  energy 
incessantly  by  communicating  its  motion  to  the  ether.  It  is 
thus  the  centre  of  a  field  of  force  and  the  lines  of  force  are 
straight  lines  radiating  from  this  centre,  while  equipo- 
tential  surfaces  are  spherical  surfaces.  The  force  at  any 

point  is ,  where  m  is  the  mass  of  the  radiating  point. 

The  integral  of  this  with  respect  to  r,  in  the  direction  of  the 
force,  is  the  work  done,  and  the  integral  in  the  opposite 
direction  is  the  work  undone  which  is  called  the  potential, 
or  stored,  energy,  represented  by  V.  Therefore  V  =  —W. 
If  we  assume  that  attraction  is  effected  through  the  agency 
of  longitudinal  waves  in  the  ether,  it  can  be  shown  (Me- 
chanics of  Electricity)  that  the  force  at  any  point  is  equal 
to  the  quantity  of  wave  energy  traversing  (orthogonally) 
unit  surface  in  unit  time.  In  other  words,  the  force  is 
measured  by  the  density  of  the  flux  of  the  energy  at  the 
point.  The  total  energy  traversing  any  surface  surrounding 
the  point  is  necessarily  the  same  as  that  crossing  any  other 
such  surface,  or  is  equal  to  the  flux  of  energy  across  unit 
sphere  about  the  point.  The  density  of  the  energy  at  any 
point  is  necessarily  inversely  as  the  surface,  or  as  the  square 
of  the  distance,  so  that  our  assumption  that  attraction  is 


46  NATURAL   PHILOSOPHY 

effected   by   longitudinal   waves   contains  implicitly  the 

result,/  = .  The  total  flux  of  energy  across  any  spherical 

surface  is  2/  =  47rm,  and  this  is  the  energy  flux  across  any 
surface  surrounding  the  point.  It  follows  that  for  any 
number  of  points,  or  for  any  distribution  of  matter,  the 
total  flux  across  any  surrounding  surface  is  ^irM.  If  the 
surface  does  not  surround  the  radiating  matter,  since  the 
energy  cannot  accumulate  within  this  surface,  but  the 
amount  of  energy  contained  by  the  surface  must  at  all 
times  be  constant,  it  follows  that  as  much  energy  is  always 
passing  out  of  the  surface  as  is  entering  it,  so  that  the  total 
flux  of  energy  across  its  surface  is  zero.  These  results 
constitute    Gauss'   Theorem.    Mathematically  expressed, 

2  4^  c/5   =  -S  -^  cos  {nr)  dS  =  0, 
an  fi  ^    ' 

when  the  matter  is  without  the  surface,  and 

2  4^  J5  =  -2  4-  cos  (nr)  dS  =  47rM, 
dn  7-2  ^     ^ 

when    the   matter    is   within   the   surface.    The   normal, 

w,  is  always  supposed  drawn  inward  from  the  surface. 

The  potential  due  to  a  particle,  w,  at  any  point  is  the 

fyi 
scalar  (undirected)  quantity,   F  =  — ,  and  the  potential 

T 

at  that  point  due  to  any  number  of  particles  is  2  — • 


19.     Poisson's  Equation 

Selecting  a  certain  point,  let  us  surround  it  with  a  sur- 
face. This  surface  may  cut  matter,  so  that  there  is  matter 
both  within  and  without  the  surface.  The  total  flux  of 
energy  through  the  surface  from  the  outside  matter  is  zero, 
while  that  from  the  inside  matter  is 

r  [^"^^  ^  ~  f  [^'"''^  ^'^^^  dS=47rM=^T  r  r  fpdr 
(1),  p  being  the  density  of  the  inside  matter  at  any  point. 


ATTRACTION   OF   A   CIRCULAR   DISC  47 

From  Green's  theorem,  putting  U  =  1, 

//^-=-///(£-r-g-r.|?)-<«. 

Designating  the  operation,  (^  +  ^  +  ^)t>yA,  and 

combining  (2)  with  (1),  we  have 

j  j  j  (aV  -\-  47rpj  dT  =  0  (3).    If  we  allow  our  surface  to 

close  down  upon  our  point,  Equa.  (3)  will  still  be  true. 
Therefore,  AV  -\-  47rp  =  0,  which  is  Poisson's  equation. 
It  expresses  the  fact  that  if  the  potential  at  any  point  due 
to  any  distribution  of  matter,  be  operated  upon  by  the 
operator  A,  the  result  will  be  —  47rp,  where  p  is  the  density 
of  the  matter  at  that  point.  If  there  is  no  matter  at  the 
point,  AF  =  0,  which  is  Laplace's  equation. 


unit  mass  situated  in  its  axis,  is    I     TT", — ZTJ  dr,  where  R 


IS 


20.     Attraction  of  a  Circular  Disc 

The  attraction  of  a  homogeneous  circular  disc  upon 

lircrar 

(r^  +  a2)i 

is  the  radius  of  the  disc,  a  the  distance  from  the  disc,  and  <r 
the  density  of  the  matter  upon  it.  Integrating, 

F  =  lira-  I  1 =^=  I  =  lira  (1  —  COS  a),  where  a 

\  ^R2  -  a2/ 

the  angle  subtended  by  R  at  the  attracted  point.  If  the  point 
is  in  the  surface,  F  =  lira.  In  crossing  the  surface,  since  the 
attraction  is  lira  on  one  side  and  —  2x0-  on  the  other  side, 
there  is  a  sudden  change,  or  discontinuity,  in  the  value  of 
F  of  4^(7 .  And  in  traversing  any  shell,  no  matter  how  mat- 
ter is  distributed  over  it,  the  change  of  the  attraction  must 
always  be  4x0-.  For  the  attraction  of  an  infinitely  small 
disc  at  the  point  will  be  2x0-,  and  that  of  the  rest  of  the 
shell  could  not  have  changed  during  the  infinitely  small 
motion  of  traversing  the  shell.  We  have  already  seen  that 
the  attraction  of  a  spherical  shell  (homogeneous)  changes 
from  47ro-  just  outside  the  shell  to  zero  within  the  shell. 


48  NATURAL   PHILOSOPHY 

The  attraction  of  an  infinitely  small  disc  of  the  shell  at 
the  point  was  lira.  Hence  the  attraction  of  the  rest  of  the 
shell  was  also  lira.  In  traversing  the  shell  the  latter  did  not 
change,  but  the  former  changed  to  —Ittct.  Hence  the 
attraction  within  the  shell  became  zero. 

21.     Maclaurin's  Theorem 

We  have  an  ellipsoid,  -^  +  fr  +  T   =   1»    uniformly 

filled  with  matter  of  a  density,  7~"("7-t-r7  +  ~)»  and 

we  wish  to  find  if  it  is  possible  to  distribute  negative  matter, 
or  matter  which  repels  instead  of  attracting,  over  its 
surface,  in  such  a  way  that  it  will  annul  the  external 
attractive  field  of  the  ellipsoid.  For  the  purposes  of  analysis 
it  would  suffice  to  make  use  of  an  imaginary  substance 
which  we  supposed  to  repel  instead  of  attracting.  Such  a 
result  would  be  obtained  by  simply  making  the  density  of 
such  a  substance  negative  instead  of  positive,  as  by  making 
such  changes  in  our  formulas  we  reverse  attractional 
actions.  It  is  certain  that  these  actions  are  effected  through 
and  by  the  ether,  and  probably  by  the  agency  of  longitu- 
dinal waves.  Assuming  this  agency,  and  calling  matter 
denser  than  the  ether,  positive  matter,  and  matter  less 
dense  than  the  ether,  negative  matter,  it  can  be  shown 
(Mechanics  of  Electricity)  that  positive  matter  attracts 
positive  matter,  while  it  repels  negative  matter.  The 
repulsions  of  nature,  of  which  there  are  many  instances 
(comets'  tails),  as  well  as  attractions,  are  probably  to  be 
explained  in  this  way,  and  in  all  these  cases  the  actual 
motions  are  accurately  expressed  analytically  by  simply 
making  the  density  negative.  The  negative  matter  which 
we  postulate  in  the  present  problem  is  therefore  not  merely 
an  imaginary  or  mathematical  conception,  but  probably 
represents  an  actual  condition  in  nature. 

But  to  return  to  our  problem.  If  such  a  distribution  of 
negative  matter  on  the  surface  of  the  ellipsoid  is  possible, 


MACLAURIN'S   THEOREM  49 

then  there  will  be  no  flux  of  energy  through  this  surface, 
and  the  potential  at  the  surface  and  at  all  external  points 
must  be  zero.  Gauss'  theorem  requires  that  the  positive 
and  negative  matter  within  the  surface  shall  be  equal,  or 
that  the  algebraical  sum  of  this  matter  shall  be  zero.  The 
potential  at  any  point,  x,  y,  0,  within  the  surface  must  be, 

1     /  X2         y2  02  \ 

V  =  y(l 2~h2 2")'^^^  ^^^^  makes  the  potential 

zero  at  the  surface,  and  at  any  interior  point  p  =  —  27-  AF, 

which  satisfies  Poisson's  equation. 

The  distribution  of  matter  over  a  surface,  without  any 
thickness,  like  the  concentration  of  matter  in  a  point,  is  a 
purely  imaginary  mathematical  conception,  but  if  we  have 
a  shell  bounded  by  two  surfaces  which  though  infinitely 
near  give  the  shell  varying  thicknesses  at  different  points, 
and  we  fill  this  shell  with  homogeneous  matter  of  any 
density  we  please,  then  we  have  a  real  distribution. 

Since  the  surface  of  the  ellipsoid  is  an  equipotential 
surface,  the  force  inward  from  this  surface  must  be  every- 
where normal,  and  this  force  is 

f:-[(S)"+(¥)"+(f)"]'=[i+s+?i]' 

Since  in  crossing  the  surface  the  force  must  change  ab- 
ruptly by  an  amount  47r(7,  where  a  is  the  density  of  the 
surface  matter  at  the  point  of  crossing,  and  since  the 
length,  p,  of  the  perpendicular  from  the  centre  to  the 
tangent  plane  at  any  point  is 


r^2  'y2         22-|-i 

[^4  +  6"4  +  ^J     'W^have 


dV  ,  1 

on  p 


Hence  if  we  distribute  negative  matter  over  the  surface 

in  such  a  way  that  at  every  point  its  density  is  —  j-  •  -, 

47r    p 

the  problem  is  solved. 

The  equation  of  an  ellipsoid  in  terms  of  the   ±,  p, 
from  the  centre  to  a  tangent  plane  at  any  point,  and 


,2 


50  NATURAL   PHILOSOPHY 

a,  jS,  7,  the  direction  angles  of  this  ±  referred  to  the 
principal  axes,  a,  b,  c,  is  p^  =  a^  cos2  a  +  62  cos2  j3  -}- 
c2  cos  27.  For  projecting  the  co-ordinates  of  the  point  on 
the  ± ,  we  have  p  =  x  cos  a  -\-  y  cos  ^  -\-  z  cos  7,  and  the 

equation  -:;  +  fr  +  —  =  1.  Varying  only  x  and  y  in  these 

Cl^  u^  C^ 

equations  we  have  dx  cos  a  -\-  dy  cos  i3  =  0,  and 

— :-  +  ^-rr  =  0.  Hence  -x  cos  /3  =  ,-3^  cos  a  with  similar 
o2  o2  a  0 

expressions  for  x  and  z,  and  y  and  s.     ^2  =  -j£;2  cos2  a  + 

);2  cos2  /8  +  22  cos2  7  +  2:J[:;y  cos  a  cos  /3  -f  2xz  cos  a  cos  7  + 

2yz  cos  /3  cos  7  =  a2  cos2  a  -\-  b^  cos2  /3  +  ^^^  cos2  7  — 

/b            ^        a             \2       /c                    a  \2 

I  -:j;  cos  p  —  -j-y  cos  o:  J    —  (  -^  cos  7 z  cos  a  j 

—  (  ry  COS  7 2  cos  j3  j 

The  last  three  terms  are  zero. 

If  we  vary  the  lengths  of  the  axes  of  an  ellipsoid  but 
keep  the  direction  of  the  ±  to  a  tangent  plane  unaltered, 
the  variation  of  the  length  of  p  will  measure  the  thickness 
at  the  point  of  tangency  of  the  shell  formed  by  the  original 
surface  and  the  varied  surface. 

cos2  a  •  6a2  +  cos2  ^  8b^  +  cos2  y  dc^ 

sp  = 2^ 

In  order  that  the  thickness  of  the  shell,  or  amount  of 

matter  in  it  at  every  point,  shall  be  proportional  to  -» 

P 
we  must  vary  the  surface  so  that  5a2  =  562  =  5^2. 

But  this  is  a  property  of  confocal  ellipsoids,  for  the 
condition  of  confocality  is  a2  —  62  =  Const.  62  _  ^2  = 
Const.,  or  5a2  =  562  =  3^2. 

Calling  the  thickness  of  the  shell  at  any  point,  t,  and 

the  common  variation  of  the  squares  of  the  semi-axes,  q, 

.  1    2t 

we  have  —  o-  =  -; 

47r   q 

If  then  we  have  an  infinitely  thin  shell,  bounded  by  two 

confocal  ellipsoids,  filled  with  negative  matter,  and  the 


THEOREM    ON   ATTRACTION  51 

inner  ellipsoid  filled  with  an  equal  amount  of  positive 
matter,  both  matters  being  homogeneous,  there  will  be  no 
external  field.  The  field  of  the  confocal  shell  is  therefore 
the  same,  but  reversed,  as  that  of  a  homogeneous  confocal 
ellipsoid  of  the  same  mass,  and  if  the  masses  are  not  the 
same  the  contours  of  the  two  fields,  or  lines  of  force,  are 
the  same,  but  the  intensities  of  the  forces  at  different 
points  are  proportional  to  the  masses. 

By  increasing  or  decreasing  our  original  ellipsoid  through 
the  addition  or  subtraction  of  infinitesimal  confocal  shells, 
but  keeping  always  the  same  amount  of  homogeneous 
matter  filling  the  ellipsoid,  we  shall  not  alter  the  external 
field.  Hence  any  confocal  shell  has  the  same  external  field 
as  any  other  confocal  shell  with  the  same  foci  and  mass, 
or  as  any  confocal  homogeneous  ellipsoid  of  the  same  mass. 
If  the  masses  are  not  equal,  the  direction  of  the  force  at  all 
external  points  is  the  same,  but  the  intensity  is  proportional 
to  the  mass.  This  is  Maclaurin's  Theorem.  Generally, 
all  thick  or  thin  homogeneous  shells  bounded  by  confocal 
ellipsoids  have  the  same  external  fields  if  their  masses  are 
equal:  if  not,  the  forces  at  any  point  are  in  the  same 
direction  and  proportional  to  the  masses. 

Since  in  the  limiting  case  of  confocal  ellipsoids,  the 
ellipsoid  becomes  an  elliptic  disc  with  semi-axes  Va2  —c^ 
and  V62  —  c2,  an  elliptic  disc  will  attract  the  same  as  any 
confocal  ellipsoid,  if  we  distribute  uniformly  upon  the  disc 
an  equal  amount  of  matter.  And  a  circular  disc  will  exert 
the  same  attraction  on  all  external  points  as  an  ellipsoid 
of  revolution  of  equal  mass,  provided  the  radius  of  the 
disc  is  Va2  —  62  and  it  is  similarly  placed. 

22.     Theorem  on  Attraction 

The  resultant  attraction  of  a  circular  disc  on  a  point  in 
its  axis  we  have  seen  to  be  2x0-  (1  —  cos  a),  (Art.  20). 
27r(l  —  cos  a)  is  the  solid  angle  subtended  by  the  disc  at 
the  point.  For  if  r  is  the  radius  of  the  sphere  which  con- 


52  NATURAL   PHILOSOPHY 

tains  the  circumference  of  the  disc  and  has  the  point  for 
its  centre,  the  surface  of  the  spherical  cap  cut  out  by  the 
disc  is  27rr.r(l  —  cos  a).  The  attraction  of  the  disc  is 
therefore  measured  by  the  solid  angle,  or  surface  of  unit 
sphere  cut  out  by  the  cone  formed  by  the  disc.  Generally 
the  resultant  attraction  of  a  homogeneous  disc  of  any 
shape  in  the  direction  OP,  the  _L  from  the  point  to  the 
disc,  is  measured  by  the  surface  cut 
out  of  unit  sphere,  drawn  about  the 
point,  by  the  cone  having  the  disc 
as  a  base  and  the  point  as  an  apex. 
An  elementary  cone  do)  cuts  out  of 
a   spherical     surface    through    A,    an 

TO 

element  dS,  and  an  element out 

cos  a 

of  the  disc.  The  resolved   part  of  the 

attraction   of   the   element  in    the    di-  ^^^-  ^• 

rection  OP  is  — -,  p  being  the  distance  of  the  element. 

p2 

S  -—  is  the  solid  angle  subtended  by  the  disc  at  0,  and  the 

p2 

theorem  is  proved.  When  the  disc  is  symmetrical  about 
the  ±,  this  measures  the  total  resultant  attraction. 
The  theorem  gives  a  means  of  finding  the  attraction  of  an 
ellipsoid  on  an  external  point,  in  the  direction  ±  to  its 
focal,  or  a,  b,  plane.  For  we  can  reduce  the  ellipsoid  to  an 
elliptic  disc  confocal  with  the  ellipsoid  and  of  equal  mass, 
and  the  attraction  of  this  disc,  ±  to  its  plane,  is  the  sur- 
face cut  out  of  unit  sphere  about  the  point  by  the  cone  of 
the  disc,  into  its  density. 

23.     Homoeoids 

For  simplicity,  we  shall  define  a  homoeoid  as  an  infinites- 
imal shell  bounded  by  two  similar  ellipsoids.  The  con- 
dition of  similarity  between  two  ellipsoids  is 

^^         I        y^         ,        2^        _  . 

a2  H+q)'^  b2  (1  +  5)  "*"  c2  (l-f  g)  "  '' 


HOMOEOIDS  53 

and  we  have   seen  that  the  condition  of   confocality  is 

X^  a;2  22 

+  7-r^. —  H — r—. —  =  1.  That  is,  the  variation  of 


a2,  62^  ^2^  is  qai  etc.,  in  the  first  case  and  q  in  the  second. 

^  =    V  a2cos2a  +  62cos2j3  +  c2cos27,  and  the  thickness,  t,  of 

a  shell  at  any  point  is  the  variation  of  the  constantly 

directed  ± ,  />,  to  a  tangent  plane  at  that  point,  as  we  vary 

the  axes  infinitesimally. 

cos2a5a2  +  cos2i3562  4-  cos275c2     ^^   .       ...       ., 

bp  =  t  -=  ■ TT- ' — .    If  the  ellipsoids 

ip 

are  similar,  bp  =  ^  =  ^  ;  if  they  are  confocal,  bp  =  t  =  ~ . 

Hence  the  density  of  matter  at  any  point  of  a  homoeoid 
varies  directly  as  p,  while  in  a  confocal  shell  it  varies 
inversely  as  /?. 

Let  us  describe  an  infinitesimal  double  cone,  Jco,  having 
some  point,  P,  in  the  interior  of  a  homoeoid  as  its  common 
apex.  These  cones  will  cut  out  elements,  dS.dp,  attracting 

the  point  with  a  force  — ir~^»  where  r  is  the  distance  of  P 

from  the  element,  p  =  r  sin  r,  where  r  is  the  angle  between 
r  and  the  tangent  plane  at  dS.    dp  =  sin  rdr,   and  the 

attraction  of  each  element  is :; =  drdoj,  since 

r2 

=  dco.    Now   any   plane  section  of  a  homoeoid 

must  be  two  similar  ellipses,  and  passing  a  plane  through 
the  axis  of  our  cones,  by  varying  its  direction,  it  will  be 
possible  to  make  this  axis  a  diameter  of  the  two  similar 
ellipses.  Hence  generally  whenever  a  straight  line  is 
passed  through  a  space  bounded  by  two  similar  and 
similarly  placed  ellipsoids,  the  two  intercepts  of  the  line 
between  the  two  surfaces  will  be  equal.  It  follows  that  there 
is  no  force  in  the  interior  of  a  homoeoid,  whether  the  shell 
is  thick  or  thin.  It  may  be  observed  that  a  charge 
of   electricity   distributes  itself   over  the   surface   of   an 


54  NATURAL   PHILOSOPHY 

ellipsoidal  conductor  homoeoidally  since  there  is  no  force 
in  the  interior,  or  the  density  of  the  electricity  at  any  point 
is  directly  proportional  to  the  ±  on  the  tangent  plane. 

A  homoeoid  is  obviously  an  equipotential  surface  for  its 
own  field,  for  the  potential  at  all  interior  points  and  at  its 
surface  is  constant.  If  we  draw  an  equipotential  surface 
just  outside  and  infinitely  near,  the  distance  between  the 
two  surfaces  will  be  inversely  as  the  density  a  at  any 

point,  for  F  "=  -f-  '  The  density  is  proportional  to  p  and  the 

thickness  of  the  shell  formed  by  the  two  surfaces  is  as  —  • 

P 
Hence  the  infinitely  near  equipotential  surface  is  an  ellip- 
soid confocal  with  the  homoeoid.  If  we  distribute  ho- 
moeoidally on  the  confocal  ellipsoid  a  mass  equal  to  the 
original  mass,  it  will  be  an  equipotential  surface  for  its 
own  field  and  the  flux  leaving  this  surface  will  be  the 
same  as  the  flux  from  the  original  mass.  Hence  if  we 
distribute  homoeoidally  equal  masses  on  two  infinitely 
near  confocal  ellipsoids  their  external  fields  will  be  the 
same.  By  a  simple  extension  of  the  reasoning  it  is  evident 
that  any  two  confocal  ellipsoids  on  which  equal  masses 
are  distributed  homoeoidally  have  identical  external  fields, 
and  if  the  masses  are  not  equal,  the  force  at  any  external 
point  is  in  the  same  direction  and  proportional  to  the  mass. 
Further,  all  external  equipotential  surfaces  of  a  homoeoidal 
distribution  are  confocal  ellipsoids.  This  is  Charles* 
Theorem. 

24.     The  Potential  Function 

Fig.  10  represents  any  two  masses  with  centres  of  in- 
ertia at  G  and  G\  P  and  P'  are  any  two  points  in  the 
masses  and  GG'  =  R,  GP  =  r,  G'P  =  p,  GP'  =  p',  and 
PP'  =  p".  The  two  bodies  have  a  mutual  field  of  energy 
represented  by  their  mutual  potential  which  for  any  two 

dyyi  dyyi 

elements  is -, — .  We  wish  to  find  the  sum  of  the  po- 

P 


THE   POTENTIAL   FUNCTION  55 

tentials  for  all  the  elements,  thus  deriving  the  total  energy 
of  the  system.  The  potential  between  the  first  body  and 

dyyt 
an  element  dm'  of  the  second  body  at  P'  is  dm'  S  -77-, 

P 
/  dm^ 

and  the  total  energy  is  S  ( dm'  S  — 77- ).  Taking  rectangular 


(,m'^p). 


co-ordinates  with  G  as  origin  and  p'  as  axis  of  x,  we  have, 
since  p"2  =  p'2  _  {2p' x  -  r^), 


^dm^^djn/2/x^y^^dm/x^^ 
P  P     \       P  2       /  P    \         P 

) 


3x2  -  r2        5^3  _  3xr2        3Sx^  -  SOx^r^  +  Sr* 

I  n     fl  I"  o  .f±  "I 


2p'2        '  2p'3  '  8p'4 


Fig.  10. 

Since  G  is  the  centre  of  inertia  all  the  odd  powers  vanish 

in  the  integration,  and  we  have 

^dm  _  ^dm  (  3x2  -  ^2        35:^^  -  30x2r2  -}-  Srj    ,\ 

p"     "         P'       V  2p'2  "^  V^  -    "^l 

S  <imr«  is  evidently  a  constant.    S  dmr2  = ;r • 

S  dmx2  =  — ^^^r Ip',  where  Ip'  is  the  moment  of 

inertia  about  p'  as  an  axis. 

The  second  term  is  therefore  x-7^ Let  us 

suppose  that  the  bodies  are  spheroids,  either  homogeneous 

or  made  up  of  homogeneous  confocal  shells.    Integrating 

we  have 

_   .      .        15  VA±B_±C       J.  ,12 


56  NATURAL   PHILOSOPHY 

In  like  manner 


S  dmx2r2  =  S  dmx^  +  ^  1"^  +  ^  +  C  _  ^n  ^^,  _^ 


2M 

35 

where  a  and  6  are  the  semi-axes  of  the  shells.  The  third 
term  therefore  contains  Ip i2,  Ipi,  Ip'^,  with  determinable 
coefficients  in  the  numerator  and  8p'*  in  the  denominator. 
Every  term  of  the  order  2«  will  have  7pi**,  /pl**"^  .  .  . 
Ip'^,  in  the  numerator  and  p'O-n  +  i)  {^  -th^  denominator. 
Hence  we  can  write 


P  P  \^    / 


where  F(  )  is  an  algebraical  series  having  the  moment  of 
inertia,  Ip,  about  an  axis  p  in  ascending  powers  in  the 
numerators,  and  p  in  ascending  powers  in  the  denominators. 
The  total  potential  is 

•'-^('-'^^)-"-'[7  +  <y)]^ 

where  I'  is  the  moment  of  inertia  of  the  second  body 
about  R  as  an  axis, 


MF{^^^^dm'F{lfy 


Here  7  and  Ip'  are  the  moments  of  inertia  of  the  first  body 
about  the  axes  R  and  p',  and  the  primes  are  the  moments 
of  inertia  of  the  second  body. 

S  dm'  F  (— r)  is  an  integral  equivalent  to  the  total 


(?) 


mass  M'  concentrated  at  some  point  in  the  second  body  at 
a  distance  p'  from  G,  into  the  average  of  the  function 


m  ■' ' 


is  obvious  that  p'  must  coincide  with  i?,  for 


THE   POTENTIAL   FUNCTION  57 

otherwise  the  forces  giving  the  translational  motions  to  the 
two  centres  of  inertia  would  not  be  equal  and  opposite 
(in  the  same  line),  and  by  their  mutuality  this  must  be 
the  case.  Hence  we  can  write 

M^       u        zr/^\       kl  +  k    ,   kl2-\-kI  -{-k   ,      , 
(1),  where  F^-^j  =  -^3—  + ^^ +  etc., 

k  representing  certain  determinable  constants.  As  has  been 
already  shown,  the  forces  acting  upon  the  first  body  are 
equivalent  to  a  single  force  acting  upon  the  mass  M  sup- 
posed concentrated  into  its  centre  of  inertia,  together  with 
a  couple  acting  about  an  axis  through  the  centre  of  inertia. 
Likewise  the  forces  acting  upon  the  second  body  are  equiv- 
alent to  a  single  force  acting  upon  the  mass  M'  concentrated 
into  its  centre  of  inertia,  together  with  a  couple  about  an  axis 
through  this  centre.  From  their  mutuality  the  two  transla- 
tional forces  are  equal  and  opposite  and  the  two  couples 
are  equal  and  opposite,  or  their  axes  are  in  opposite 
directions,  since  a  couple  is  represented  by  a  vector. 
The  force  in  any  direction  is  the  rate  at  which  the  potential 
energy  is  used  up  in  that  direction,  or  the  derivative  of  the 
potential  with  respect  to  the  direction.  The  force  urging 

the  centres  of  inertia  together  is  -j^  •  If  we  simply  turn  the 

ciK 

bodies  about  R,  the  potential  is  not  altered. 

Considering  the  second  body  simply  as  a  material  point 

of  mass  M',  Equa.  (1)  becomes 

V  =  ^  +  M'f(|).  (2) 


R 


It  will  be  noted  that  the  attraction  in  the  plane  of  the 
equator  of  a  spheroid  is  greater  than  if  the  mass  were 
concentrated  into  its  centre,  while  the  attraction  in  its 
axis  is  less,  or  for  equal  distances  the  attraction  is  a 
maximum  in  the  plane  of  the  equator  and  a  minimum  in 
the  axis .  It  will  also  be  noted  that  the  motion  of  the  centre 


58 


NATURAL   PHILOSOPHY 


of  inertia  is  the  same  whether  we  apply  the  forces  parallel 
to  their  original  directions  to  this  centre,  or  subject  the 
mass  concentrated  into  this  centre  to  the  field.  In  general 
this  is  not  the  case.  [See  Art.  8.] 

If  7  is  the  angle  which  R  makes  with  the  equator  of  the 
spheroid,  then  the  couple  tending  to  bring  the  equatorial 

plane   into    coincidence  with    K   is  -r-   =  —  -ttftt  -r   = 

ay  2R^  ay 

-^  {C  —  A)  sin  7  cos  y,  since  I  =  C  COS27  +  A  sin2  7, 

to  a  close  approximation. 


25.     Rotary  Motion 

A  disc  Spins  (Fig.  11)  on  its  axis  PP\  which  is  held  in  a 
ring  which  can  turn  about  a  horizontal  axis  HH\  and  the 

whole  turns  about  a  vertical 
axis  VV. 

If  the  disc  is  not  rotating 
and  we  turn  it  about  the  ver- 
tical axis,  it  will  simply  obey 
the  turn   and  its  axis   PP' 
will  remain  in  the  horizontal 
plane.    If  it  is  rotating  and 
we  turn  the  axis  through  a 
horizontal   angle   d\l/  in  the 
time   dt,    the   axis   will   not 
remain  horizontal.  Let  a>  be 
the    angular    velocity    with 
which  the   disc   spins.   This 
velocity  cannot  be  influenced 
by  the  turn  since  there  is  no 
couple  about    PP',  and  we 
further  suppose  no  friction. 
Let  C  be  the  moment  of  inertia  about  the  axis  of  the 
disc,  and  A  that  about  an  axis  in  its  plane.  We  can  resolve 
the   original   moment   of  momentum,  Ceo,  into   the  two 
components  Ceo  cos  dyp  and  Ceo  sin  dyp  in  the  horizontal 


Fig.  11. 


ROTARY   MOTION 


59 


plane.  Since  no  motion  can  cease  instantly,  when  we  turn 
the  axis  of  the  disc  through  the  angle  <i^  in  the  infinitesimal 
time,  dt,  the  above  moments  will  persist,  and  the  moment 
of  momentum  about  the  axis  will  be  Ceo  cos  d\l/  and  there 
will  also  be  a  moment  Ceo  sin  d\f/ 
about  a  ±  axis  in  the  horizon- 
tal plane.  The  rate  of  change  of 
a  moment  of  momentum  meas- 
ures a  couple  about  its  axis.  The 
rate  of  change  of  the  moment 
about  the  axis  of  the  disc  is 
Cu)  cos  dxp  —  Coi 


dt 


and  the  rate 


of  change  about  the  ±  axis  is 

^    ,, .  At  the  limit  the  first 

at 

becomes  zero,  and  the  second  is 
Cco^.  Hence  at  the  beginning  of 
the  turn  a  couple  is  set  up  tend- 
ing to  bring  the  axis  of  the  disc 
into  coincidence  with  the  axis  of  the  turn,  and  if  the  turn  is 
continued  these  two  axes  will  coincide  both  in  direction 
and  sense  of  motion  about  them.  Generally,  whenever  a 
rotating  mass  is  turned  about  an  axis  ±  to  its  rotation 
axis,  a  couple  is  set  up  about  an  axis  _L  to  the  former  two 
axes,  and  this  is  called  the  gyroscopic  couple.  Any  rotating 
mass  is  a  gyroscope.  If  t?  is  the  angular  acceleration  about 
the  gyroscopic  axis,  Ccoxl/  =  —A^  (1).  This  is  the  funda- 
mental gyroscopic  law  from  which  all  the  properties  of 
rotary  motion  can  be  derived.  The  gyroscopic  couple  is  an 
internal  force  due  to  the  inertia  of  the  particles  of  the  disc 
and  is  nothing  more  than  the  moments  of  the  centrifugal 
forces  which  arise  during  the  turn. 

In  Fig.  12  a  lamina  can  turn  about  a  horizontal  axis 
HH^  and  this  axis  is  turned  about  a  vertical  axis,  VV\ 
with  angular  velocity  xp.  A  point  P  has  co-ordinates,  x,  y, 
referred  to  axes  through  the  centre  of  the  lamina,  and  the 


60 


NATURAL   PHILOSOPHY 


radius  r  makes  an  angle  ^  with  the  axis  of  %.  The  cen- 
trifugal force  of  an  element  dm  at  P,  is  dmr  sin  (t?  —  ^)^2. 
The  moment  of  this  force  about  the  horizontal  axis  is 
dmr  sin  (t?  —  ip)  cos  (t?  —  (p)  xj/^. 


Since 


cos  (p  and 


=  sm  (p, 


X 

r  r 

the  total  moment  is 

Sc^m  sin  t?  cos  t?  {x^  —  y'^)p  —  SJm  (cos2t?  --  s\n^d)xyp. 
From  symmetry,  ll,dmxy  vanishes,  and  we  have  the  well 
known  result, 

2<im  sin  t?  cos  t?  {x'2-  —  y2)p  =  (^Q  —  A)  sin  ■&  cos  t?i/'2, 
where  C  is  the  moment  about  the  y  axis  and  A  that  about 
the  X  axis. 

But  we  can  get  this  result  at  once  by  gyroscopic  prin- 
ciples. Resolving  ^  into  an  angular  velocity  yj/  sin  ??  about 
the  y  axis  and  yj/  cos  t?  about  the  x  axis,  we  see  that  these  are 
two  gyroscopic  couples  about  HH\  viz.,  C  sin  ^yp.  cos  ^4/ 
and  A^  cos  ^.  ^  sin  ??,  opposing  each  other.  The  net  result 
is  {C  —  A)  sin  t?  cos  t?t/'2,  tending  to  place  the  x  axis 
horizontally.  The  gyroscopic  couple  is  therefore  merely  the 

-  integral    of    all    the 

^^^^^^^  f^  centrifugal  moments. 

j        *^\P*       ^^    another    ex- 
ample let  us  suppose 
a  sphere,  Fig.  13,  ro- 
tating about  an  axis 
PP'  with  angular  ve- 
locity  CO,    while  PP' 
can    turn    about     a 
horizontal  axis  HH', 
which    is     held     by 
a   fork   which   turns 
about  a  fixed  centre 
at   5  in   a   horizontal   plane,    with   angular   velocity   yp. 
Resolve  the  rotation  co  into  co  cos  t?  about  a  vertical  axis 
and  03  sin  t?  about  SO.  Let  SO  =  D.  Considering  a  ring  of 
matter  about  SO,  the  horizontal  velocity  of  any  particle 


Fig. 


EULER'S   DYNAMICAL   EQUATIONS  61 

in  this  ring  is  D^j/  —  rco  sin  t?  cos  <^,  where  <p  is  the  angle 
any  radius  makes  with  the  vertical.  The  centrifugal  force 

is  -jT-  [Dxp  —  ro)  sin  ??  cos  <^]2  and  the  moment  of  this  force 

dm  r   •  n^ 

about  HH^  is  -j=r     Dip  —  roy  sin  d^  cos  (p      r  cos  cp.  The 

% 
sum  of   the  moments  is    I  —    Dxj/  —  rco   sin  t>  cos    cp 

0 

r  cos  (pdcp  =  lirr.  r'^oi  sin  t?i/'  =  mr2co  sin  t?i/'.  The  moment 
of  momentum  of  the  ring  about  SO  is  mr'^oi  sin  t?  and  \j/  is 
the  angular  velocity  with  which  the  momental  axis  turns. 
Hence  the  integral  of  the  centrifugal  moments  about 
HH'  is  simply  the  gyroscopic  couple. 
For  a  disc  the  total  centrifugal  moments  about  HH' 

are  S  Itt^cjj  sin  t?^Jr  =  -77—-  w  sin  ^xp.  But  7ri?2  is  the  mass 
0  2 

i?2 

of  the  disc  and  -^  is  ^2^  where  k  is  the  radius  of  gyration, 

and  the  integral  gives  simply  the  gyroscopic  couple. 

For  the  whole  sphere,  using  the  relation  r^  =  R2  —  k^, 
where  k  is  the  distance  of  any  vertical  disc  from  the  centre, 

STTf*  CO  sm  t?^  dk  =  — :z —  — =-  •  o)  sm  ^xp.  — ^—  is  the  mass 

2R2 

and  — ^  =  k^,  and  we  see  again  the  identity  of  the  gyro- 
scopic couple  with  the  sum  of  the  centrifugal  moments. 
The  gyroscopic  principle  gives  at  once  the  total  centrifugal 
moments  acting  about  any  axis.  Without  its  use  many 
rotational  problems  would  be  practically  insoluble. 

26.     Euler's  Dynamical  Equations 

Let  us  suppose  that  a  triaxial  body  is  given  an  impulsive 
velocity  about  some  axis  through  its  centre  of  inertia, 
which  is  fixed.  The  resulting  motion  will  in  general  be 
unstable.  For  we  can  resolve  the  impulsive  velocity  into 


62  NATURAL   PHILOSOPHY 

coi,  0)2,  0)3  about  the  three  principal  axes  and  it  is  evident 
that  a  pair  of  gyroscopic  couples  will  result  about  each  axis. 
With  due  regard  to  signs,  we  can  write 

{B  —  C)  0)2003    =   Acoi 

{C  —  A)  C01C03  =  Bu)2 

{A    —    B)  0)10)2    =    Coii 

These  are  Euler's  dynamical  equations.  The  mutual 
interaction  of  these  couples  will  cause  the  original  in- 
staneous  axis  to  shift  continually. 

If,  however,  the  original  velocity  were  imparted  about 
a  principal  axis,  the  motion  would  be  stable,  for  there 
would  be  no  couples. 

Let  us  suppose  a  uniaxial  body  to  be  rotating  about 
some  axis  which  is  struck  a  sharp  blow  in  any  direction. 
We  always  consider  the  centre  of  inertia  as  fixed.  If  the 
body  were  not  rotating  it  would  simply  turn  about  an 
axis  ±  to  the  blow.  But  rotating,  the  instantaneous 
velocity  imparted  by  'the  blow  combines  with  the  original 
rotation,  and  we  have  a  new  rotation  about  an  axis  which 
is  in  the  plane  of  the  other  two.  The  rotation  axis,  instead 
of  moving  in  the  direction  of  the  blow,  in  fact  moves  in  a 
direction  ±  to  it.  If  ^  be  the  impulsive  velocity  imparted 
about  an  axis  ±  to  the  rotation  axis  and  i  the  angle  the 

new  resultant  axis  makes  with  the  original  axis,  tan  i  =  — , 


and  the  resultant  angular  velocity  is  V  ^^  +  w^-  Since  all 
axes  are  principal  axes,  the  new  rotation  will  be  stable. 

27.     Biaxial  Bodies  Under  No  Forces 

If  we  subject  a  biaxial  body  to  an  impulsive  couple, 
measured  by  G^  about  an  axis  making  an  angle  t?  with  the 
C  axis,  or  axis  of  unequal  moment,  this  is  equivalent  to  an 
impulsive  moment  about  the  C  axis,  together  with  an 
impulsive  component  about  a  ±  axis  in  the  GC  plane. 
The  instantaneous  axis  will  lie  in  this  plane.  If  o),-  is  the 
instantaneous  angular  velocity  and  i  the  angle  which  the 
instantaneous  axis  makes  with  C,  while  t?  is  the  angle  G 


TRIAXIAL   BODIES   UNDER    NO   FORCES  63 

makes  with  C,  G  cos  t?  =  Cco^  cos  i,  and  G  sin  t?  =  Acoi  sin  ^. 
Hence  A  tan  i  =  C  tan  t?  (1),  and  t  and  to,-  are  determined. 
G  is  represented  by  a  vector,  constant  in  amount  and 
direction,  and  is  called  the  Invariable  Line.  Since  there 
can  be  no  rotation  or  moment  about  an  axis  _L  to  G, 
the  C  axis  and  the  instantaneous  axis,  which  always  lie 
in  the  GO  plane,  cannot  change  their  inclinations  to  the 
invariable  line.  The  motion  therefore  will  consist  of  a 
rotation  of  the  GCi  plane  around  the  invariable  line. 
Let  yp  be  the  angular  velocity  of  this  plane.  Then  the  C 
axis  turns  about  a  ±  axis  in  this  plane  with  angular 
velocity  yp  sm  d^  =  oji  sin  i,  and  the  angular  velocity  about 
the  C  axis  is  coj  cos  t  =  co.  It  is  readily  seen  that  the  gyroscopic 
couples  about  an  axis  J_  to  the  GO  plane  exactly  balance, 
so  that  there  can  be  no  motion  about  such  an  axis.  For 
the  gyroscopic  couples  about  this  axis  are  CujxI/  sin  t?  and 
Ayp  sin  t?.  ^  cos  ^,  and  by  (1)  these  are  equal  and  opposite. 
The  motion  is  thus  completely  determined.  It  consists  of  a 
steady  rotation  of  the  CGi  plane  about  the  invariable  line, 
which  is  the  axis  of  the  impulsive  couple,  and  the  unequal, 
or  C,  axis,  and  the  instantaneous  axis  are  fixed  in  this 
plane.  G,  A,  C  and  t?  are  given,  and  from  these  w,  w,-,  i 
and  ^  are  readily  found.  The  motion  of  the  unequal  axis 
about  the  invariable  line  is  called  the  Precession,  and  xp 
is  the  precessional  velocity. 

28.     Triaxial  Body  Under  No  Forces 

When  a  triaxial  body  is  subjected  to  an  impulsive 
couple  about  an  axis  through  its  centre  of  inertia,  the  case 
becomes  more  complicated,  li  A,  B,  C  are  the  principal 
moments  of  inertia  of  such  a  body,  in  ascending  order,  then 
Ax^  -\-  By^  +  Cz^  =  1  is  the  momental  ellipsoid  of  the 
body,  and  we  have  seen  that  it  has  the  property  that  the 
square  of  the  reciprocal  of  any  of  its  radii  is  equal  to  the 
moment  of  inertia  of  the  body  about  that  radius. 

Draw  a  radius,  r,  to  any  point  on  the  momental  ellip- 
soid and  on  the  plane  tangent  at  this  point  drop  a  ± ,  ^, 


64  NATURAL   PHILOSOPHY 

from  the  centre.  The  equation  of  the  momental  ellipsoid  is 

S    +   S    +   S    =   ^^'  +  ^y  +  CZ2=1  (1) 

Whence  ^  =  -  -{•  ^  +  -  =  Aixi  +  B'^y^  -^  C^z^^ 
p2        a*        0*        c^ 

r2  (^2  cos2ai  +  52  cos2i3i  +  C^  COS27O  (2),  whereat  /3i,  7I 
are  the  direction  angles  of  a  radius  referred  to  the  principal 
axes.  If  now  we  apply  an  impulsive  couple,  G,  having  p 
as  its  axis,  the  moment  of  momentum  of  the  body  about 
this  particular  p  must  remain  constant  throughout  the 
motion  and  equal  to  G.  Resolved  into  its  components  about 
the  principal  axes,  G  cos  a  =  Aon,  G  cos  jS  =  Boii,  G  cos  7 
=  Ca)3,  where  a,  /3,  7  are  the  direction  angles  of  p  referred 
to  the  principal  axes,  and  coi,  W2,  W3  are  the  angular  veloci- 
ties about  these  axes  at  any  instant.  Hence,  G^  =  A^ooi^  -f- 
52^22  +  C2CO32  (3).  Also,  since  the  kinetic  energy,  T,  must 
remain  constant,   Acot^  +  Bo)2'2  +  CC032   =  2T  —  Ico,^  = 

-^  (4),  where  I  and  co,-  are  the  moment  of  inertia  and  the 

f2 

angular  velocity  about  the  instantaneous  axis.  Hence 
coj  =  V  2T.r,  or  the  instantaneous  velocity  is  proportional 
to  the  radius  of  the  instantaneous  axis. 

Smce  —   =  cos  as  —   =  cos/3i,  —   =  cos  7^  (5) 

coj  CO,-  a)j 

.  ^  Wi2       1  A2cOl2  +  52co22  +  C2CO32  G^       , ,, 

^^^        ''  =   2T>   =  2T =   27    (^) 

Hence  the  end  of  the  instantaneous  axis  will  always  be  in 
the  original  tangent  plane,  which  is  therefore  fixed  and 
called  the  Invariable  Plane.  The  perpendicular,  p,  is 
constant  in  length  and  direction  and  is  called  the  Invari- 
able Line.  The  angular  velocity  of  the  body  about  p  is 

p        27 
coj  —  =  -7;-,  and  it  is  therefore  constant.  If,  then,  we  sup- 

r  Cr 

pose  the  momental  ellipsoid  to  roll  on  the  invariable  plane 
preserving  a  constant  angular  velocity  about  p,  the  motion 
of  the  body  will  be  exactly  represented. 

The  invariable  line  and  the  instantaneous  axis  cut  out 


TRIAXIAL   BODIES   UNDER    NO   FORCES  65 

cones  in  the  body  during  the  motion,  and  these  are  called 
the  invariable  and  instantaneous  cones.  Their  equations 
are  found  from  (3)  and  (4),  which  combined  are 

(i4cOi2   +  Bc022   +  Cu)32)    G2    =    (A2cOl2   +  B^OJz^  +  €203^2)   2T . 

Taking  co-ordinates,  x,  y,  z,  proportional  to  the  direction 
cosines  of  either  the  invariable  line  or  the  instantaneous 
axis,  and  eliminating  coi,  co2,  £03  by  the  relations  G  cos  a  = 
Ao)i  =  kGx,  etc.,  for  the  invariable  line,  and  by  the  relations 

X  =  ka\  =  k  —  ,  etc.,  for  the  instantaneous  axis,  we  have  as 

CO,- 

the  equations  of  the  invariable  and  instantaneous  cones 
respectively, 
2AT  -  G2    ^    ,    2BT  -  G2    ^   ,    2CT  -  G2    ^       ^     .^^ 

~~A ^   "^  B ^    ^  C ^    ^  ^    ^^^ 

A  {2AT-G2)  x2+B  (2BT-G2)  y2-\-C  {2CT-G2) 02  =0  (8) 
When  2AT,  or  2CT,  equals  G2,  these  cones,  which  are  of 
the  second  degree,  become  two  imaginary  planes,  which 
however  intersect  in  a  real  line.  From  (1)  and  (5)  this  con- 
dition is  that  p  =  a,  or  p  =  c,  or  the  instantaneous  axis 
coincides  with  the  major  or  the  minor  axis  of  the  momental 
ellipsoid  at  the  beginning  of  the  motion.  If  2BT  =  G2, 
the  cones  reduce  to  two  real  planes.  Here  p  =  6,  or  the  ± 
is  equal  to  the  middle  axis  of  the  ellipsoid. 

The  instantaneous  cone  intersects  the  surface  of  the 
ellipsoid  in  a  curve  which  is  called  the  Polhode.  This  curve 
is  evidently  the  locus  of  all  those  points  on  the  ellipsoid  for 
which  p  has  a  constant  value.  Its  equation  is  found  by 
combining  the  equation  of  the  ellipsoid  with  another  ex- 
pressing the  fact  that  p  remains  constant.   Hence  it  is 

1  G2 

since  —    =  A2^2  +  ^2^2  +  c:222  =  --. 

p2  Z 1 

This  is  the  equation  of  a  cone  of  the  2nd  degree  with  its 
apex  at  the  centre.  If  p  is  equal  to  the  middle  axis,  or  —  =  B, 
we  have  A  (B  —  A)  x2  =  C  {C  —  B)  z2,  which  represents 


66 


NATURAL   PHILOSOPHY 


two  planes  intersecting  in  the  middle  axis  and  making 
angles  with  the  A,  B,  plane  whose  tangents  are 


i 


A  {B  -  A) 


Taking  the  equations, 


C  (C 

1 


B) 


G2 


=   A2X2   +   B2y2   +  C2Z2    =    ^^ 


and  Ax2  -\-  By^  +  Cz^  =  1,  and  eliminating  y,  we  have  as 
the  projection  of  the  polhode  on  the  AC  plane, 


A{B  -  A)x2  -  C  {C  -  B)  02 


These  projections  are  therefore  hyperbolas.  In  like  man- 
ner we  see  that  the  projections  upon  the  AB  and  BC 
planes  are  all  ellipses. 

The  case  is  peculiar  for  G^-  =  2BT,  or  p  =  b.  All  the 
polhodes  are  projected  upon  the  AC  plane  as  hyperbolas, 

but  in  this  case  the  hyperbolas 
reduce  to  two  straight  lines  in- 
tersecting in  the  centre.  The 
polhodes  in  this  case  become 
two  ellipses  and  they  are  called 
the  separating  polhodes,  be- 
cause all  the  polhodes  on  one 
side  of  them  enclose  the  major 
axis,  while  all  the  polhodes  on 
the  other  side  enclose  the  minor 
axis.  In  Fig.  14  these  polhodes 
and  various  other  polhodes  are 
shown. 

Analyzing  the  motion  for  this 

special    case   we    get    the    fol- 

From    the    fundamental    equations    (3) 


lowing  results: 
and  (4), 

C0l2 


0)3^ 


c  - 

-  B 

c  - 

-A 

B  - 

-  A 

Q2    -   B2CC2^ 

AB 

Ql    -    B2032^ 

BC 


and 


C02 


C  -  A 
B 


£010)3. 


TRIAXIAL   BODIES   UNDER   NO   FORCES 


67 


Putting 


V 


{B  -  A)  {C  -  B) 


AC 


k, 


dc02 


G2 
B2 


kdt. 


TTr,    ~~    W2^ 


Integrating, 


G  +  Boi2 


=  Ee 


G   —  B(j02 

where  £  is  a  determinable  constant. 
Since  G  cos  jS  =  ^0x2,  this  becomes 


1  +  cos  /3 


a 

=  ctn^  T.  =  Ee 


IkGt 


2kGt 
B 


(9). 


1  —  cos  ^ 

From  (7)  and  (8),  the  invariable  and  instantaneous 
cones  become  in  this  case  two  planes  intersecting  in  the 
middle  axis  and  making  angles  with  the  AB  plane  whose 
tangents  are  respectively 


i 


B  -  A 


and  =t= 


i 


B 


C  -  B IC    C  -  B 

In  Fig.  15,  B,  p,  and  I  are  the  points  respectively  where 
the  middle  axis,  the  invariable  line  and  the  instantaneous 
axis   pierce    a   unit  _ 

sphere    about     the 
centre.   As   the   in- 
variable    line     de- 
scribes its  plane  in 
the  body,    the  arc 
Ip  must  always  be 
JL  to   the  arc  Bp, 
and    the    angle    B 
between  the  invari- 
able  and   instanta- 
neous  planes  is   constant.    The  right   spherical   triangle 
B  p  I  therefore  always  remains  similar,  and  the  problem  is 
reduced  to  determining  the  cones  described  in  space  by 
the  corners,  B  and  I,  of  this  triangle  as  it  rotates  about  the 

invariable  line  with  constant  angular  velocity,  •^-  From  (9) 

it  is  evident  that  as  the  time  increases  the  angle  /3  ap- 
proaches the  value  zero  or  tt.  The  middle  axis  therefore 


Fig.  15. 


68  NATURAL   PHILOSOPHY 

moves  so  as  to  place  itself  in  coincidence  with  the  in- 
variable line,  the  direction  of  the  motion  being  such  that 
like  poles,  or  like  rotations,  coalesce.  The  triangle  B  p  I, 
always  remaining  similar,  finally  is  reduced  to  nothing 
and  the  motion  becomes  a  steady  one  about  the  middle 
axis. 

Differentiating  (9),  it  is  seen  that  the  linear  velocity 
along  a  meridian,  /3,  p  being  the  pole,  is  proportional  to 
sin  /3,  and  the  linear  velocity  ±  to  this,  or  along  a  parallel 
of  latitude,  is  likewise  proportional  to  sin  0,  since  the  angu- 
lar velocity  about  p  is  constant.  Hence  the  middle  and 
instantaneous  axes,  as  they  spiral  inward  towards  p,  cut 
every  meridian  at  a  constant  angle,  and  the  paths  are 
what  are  called  rhumb  lines. 

The  polhode  as  it  rolls  on  the  invariable  plane  traces 
out  on  this  plane  a  curve  called  the  Herpolhode.  Its  general 
character  can  be  seen  from  Fig.  15.  It  is  limited  by  two 
circles  which  it  alternately  touches  and  it  is  symmetrical 
about  points  of  tangency.  For  the  special  case  where  the 
instantaneous  axis  is  in  the  separating  polhode,  the 
herpolhode  is  quite  different.  It  is  shown  by  the  oval  in 
Fig.  15.  If  the  directions  of  rotation  about  the  middle  axis 
and  p  are  similar,  the  path  curves  sharply  towards  p  where 
at  an  infinitely  small  distance  it  continues  to  approach  p 
indefinitely.  If  the  rotations  about  the  middle  axis  and  p 
are  in  a  contrary  sense,  the  instantaneous  axis  moves  in  the 
herpolhode  at  first  away  from  p  and  then,  the  body 
turning  over,  curves  around  sharply  from  the  other 
direction  towards  p,  where  as  before,  from  an  infinitely 
small  distance,  it  approaches  the  pole  indefinitely.  The 
instantaneous  axis  moves  to  the  pole  practically  in  a  little 
less,  or  a  little  more,  than  a  quarter  of  a  turn,  but  since  it 
cannot  move  directly  to  the  pole  and  then  stop  abruptly, 
it  moves  first  to  an  infinitely  small  distance  from  the  pole 
and  then  continues  its  approach  indefinitely. 


GYROSCOPES   UNDER   EXTERNAL   FORCES 


69 


29.   Gyroscopes  under  External  Forces 

Any  rotating  mass  is  by  definition  a  Gyroscope.  Let  us 
suppose  a  top,  Fig.  16,  rotating  about  its  axis  with  angular 
velocity  co  and  held  at  an  angle  ??<,  to  the  vertical.  It  is 
then  abandoned  to  gravity  and  we  wish  to  find  the  motion. 
The  centre  of  inertia,  G,  is  at  a  distance,  h,  from  the  point 
of  support.  The  moment  of  in- 
ertia about  its  axis  is  C  and 
that  about  a  ±  axis  through  0, 
A.  We  shall  take  as  axes  of 
reference,  OG,  an  axis  ±  to  this 
in  the  vertical  plane  through  0, 
and  a  horizontal  axis  through 
0  _L  to  the  other  two.  If  yj/  is  ^^ 
the  angular  velocity  of  the  plane 
GOV  about  OF,  the  top  turns 
about  the  i/'  sin  d^  axis  with 
angular  velocity  ^  sin  t>  and 
this  axis  turns  about  the  OG  axis  with  angular  velocity  yp 
cos  d^.  We  have  then  as  the  equations  of  motion 

mgh  sin  d^  —  Ccxjxp  sin  ??  +  Axl/^  sin  t?  cos  t?  =  A^     (1) 
Ccoi}  -  ArP  cos  M  =  ADt  (tp  sin  ??).  (2) 

Integrating  (2),  Ceo  (cos  ^o  —  cos  ^)  =  A\p  sin2  ^,    (3) 
Multiplying  (1)  by  4,  (2)  by  rp  sin  t?,  adding  and  integrating 


Fig.  16. 


Wg/t  (cos  t>o 


,,  At?2         A^2  sin2  z? 

cos^)  =— + 2 • 


(4) 


Equa.  (3)  merely  states  that  the  moment  of  momentum 
about  OV  remains  constant,  while  Equa.  (4)  states  that 
the  increase  of  the  kinetic  energy  is  equal  to  the  work  done 
—  both  of  which  were  a  priori  evident.  These  equations 
determine  the  motion  completely. 

Let  us  now  find  the  path  (guided  if  necessary)  which  the 
axis  of  the  top  must  take  in  order  that  it  may  move  from 
some  point  1  in  its  actual  path  to  any  other  point  2  in  its 
actual  path,  in  the  shortest  time  possible. 


70  NATURAL   PHILOSOPHY 

ds 


and 


Since  ^2gh  (cos  t?o  ~  cos  t?)  =  -7^,  where  ds  is  an  element 

of  path  described  by  the  extremity  of  the  radius  of  gyration 
k,  where  mk^  =  A, 

dt  =  ^"  , 

^2gh  (cos  ??o  -  cos  ?>) 

p ^£ 

ij     V2g/t  (cos  t?o  —  cos  t?) 

Calling  the  angle  which  this  path  makes  with  a  meridian 

at  any  point,  r,  ds  =  kdd-  sec  r.  We  shall  take  ??  as  the 

independent  variable  and  vary  s.  5ds  =  kdd^  sec  r  tan  rbr. 

„.         s\nM\l/       ^  ^^  ^    ^         sint?^,, 

Since  — T7 —  =  tan  r,  5  tan  r  =  sec2  tBt  =  —jir  5a^^. 
an  av 

sin  z>  ddyp      .._.  ,  .         ,     .      „    . 

5t  =  —Tz z— '  Hence  8ds  =  k  sm  t?  sm  ra5^ 

at?    sec2  T  ^ 

1  ►T'  r^       k  sin  t?  sin  T(i5iA 

and  8T         ^  ^ 


-J 

Integrating,  dT  = 


yl2gh  (cos  t?o  —  cos  t?) 
fe  sin  z?  sin  r  5r/'        |2 


V2g/i  (cos  ^o  -  cos  t?)  |i 
k  sin  t?  sin  r 


S'H 


^2gh  (cos  t?o  —  cost?)/ 
Since  the  limits  are  fixed  the  first  term  vanishes  and  the 
condition  that  the  path  shall  be  the  one  of  quickest  motion 

k  sin  t?  sin  r 

^2gh  (cos  t?o  —  cos  t?) 
where  K  is  an  arbitrary  constant  —  say  ^r — -.    Hence 

sin  t?  sin  r  =  -^  J  cos  t?,  -  c]^  (5) 

VA  l  2mg/j 

is  the  equation  of  the  curve  in  terms  of  the  co-ordinates 
t?  and  T.  Substituting  for  sin  r, 

k  sin  t?(j^ k  sin  t?;/' 

ds        ~  V2g/t  (cos  t?o  -  cos  t?)* 
we  have  A  sin2  t?^  =  Ceo  (cos  t?o  —  cos  t?) .  This  identifies 
the  curve  with  the  actual  gyroscopic  path.  Hence  the  axis 
of  the  top  moves  naturally  from  one  point  to  another  of  its 


GYROSCOPES   UNDER   EXTERNAL   FORCES  71 

path  in  the  quickest  possible  time,  and  generally  on  a 
spherical  surface  a  body  under  the  influence  of  gravity 
moves  from  one  point  to  another  in  the  least  possible 
time  if  it  takes  a  gyroscopic  path.  If  the  spherical  surface 
becomes  infinitely  large  these  gyroscopic  paths  become 
plane  cycloids,  for  sin  t?  becomes  constant  and  the  meri- 
dians become  parallel  vertical  lines,  while  cos  z?o  —  cos  t> 
measures  y  from  cos  i^o-  The  rectangular  equations  of  a 
cycloid  are  x  =  a  (z?  —  sin  t?),  >'  =  a  (1  —  cos  t?). 

Sin  r  = -=i^=  =  Vf  • 

or  2a  sin2  r  =  ^^  is  the  equation  of  a  plane  cycloid  in 
terms  of  r  and  y. 

We  have  already  used  this  equation  in  Art.  4,  where  we 
found  that  the  cycloid  is  the  curve  of  quickest  passage  for 
a  plane  surface.  We  have  now  found  that  the  gyroscopic 
path  is  the  curve  of  quickest  passage  for  a  spherical  surface. 

Taking  the  equation  of  the  gyroscopic  path, 

.     „   .  WW    ^ ,  cos  t?o  —  cos  t? 

sm  t7  sm  r 


^lA    1  2mgl 

since  at  the  beginning  sin  r  =  0,  the  path  is  at  first  ver- 
tically downward.  When  sin  r  =  1  the  path  is  horizontal 
and  this  marks  the  maximum  fall.  We  have  for  this  point 

cos  t?    =      ^^''^'  ^/l    -    (Cco)2cos\?o     .  (C^ 


-VI 


4mghA         1  2mghA  \6m'^g^h'^A'^ 

There  is  another  value  with  +  between  the  terms,  but 
this  is  inadmissible  since  it  makes  the  value  of  cos  t> 
greater  than  unity.  It  will  be  noted  that  at  this  point  the 
gyroscopic  couple  tending  to  raise  the  top  is  just  twice  the 
gravitational  couple.  After  this  the  axis  rises  symmetrically 
to  its  original  height  where  it  is  momentarily  at  rest  and 
then  repeats  the  process  indefinitely.  The  path  is  like  a 
series  of  festoons  hung  upon  a  parallel  of  latitude  at 
equidistant  points.  If  we  suspend  the  gyroscope  so  that 
it  can  move  freely  below  the  point  of  suspension  we  have  a 
gyroscopic  pendulum.  The  axis  executes  a  festoon  motion, 


72  NATURAL   PHILOSOPHY 

but  will  never  be  in  the  nadir  as  long  as  it  has  the  least 
rotation.  When  the  rotation  ceases  there  is  only  a  single 
festoon  hung  at  points  180°  apart  and  this  passes  through 
the  nadir.  From  (4),  we  have  in  this  case 

t  =    f  ^^  d^ 

J  ^Imgk  (cos  d^o  —  cos  t>) ' 
which,  as  we  have  seen,  is  the  law  of  the  ordinary  pendulum. 
The  motion  about  the  vertical  axis  is  the  precession, 
while  the  motion  along  a  meridian  is  the  nutation.  When  w 
is  large,  the  amplitude  of  the  vertical  vibrations  is  very 
small  and  the  vibrations  are  very  rapid,  so  that  in  high 
spinning  gyroscopes  (tops)  the  eye  cannot  detect  them,  or 
at  most  only  a  slight  blurring.  But  the  ear  can  hear  these 
vibrations  and  that  is  the  cause  of  humming  in  tops.  The 
rapidity  of  the  vibrations  can  be  measured  by  the  note. 

With  a  slight  nutation  the  festoons  become  very  nearly 
minute  cycloids,  for  making  these  small  relatively  to  the 
surface  is  the  same  as  making  the  surface  very  large 
relatively  to  the  festoons,  and  in  either  case  these  become 
plane  cycloids. 

Otherwise,  since  when  ^  and  d^  become  very  small  we  can 
neglect  their  squares  and  products,  the  equations  of  motion 
become  mgh  sin  t?  —  Coiyp  sin  t?  =  At? 

Coi^  =  ADt  ()//sin  ^). 
Taking  rectangular  co-ordinates  at  the  point  of  rest, 
mgh  sin  ^  —  Ccox  =  Ay 
Cojy  =  Ax. 
Integrating 

mgh  sin  t?  A  VCcat         .     /Ceo  A  "I 

*  =  — c^^^r-  [—  -  ^'"  [-a)\ 

mgh  sin 


y  = 


C2w2 

These  are  the  equations  of  a  cycloid  generated  by  a 
circle  of  radius     ^^  ^ >  rolling  with  uniform  angular 

velocity,  —p ,  below  a  parallel  of  latitude.  The  time  of  a 


DRIFT   OF   RIFLED   PROJECTILES 


73 


Fig.  17. 


complete  precession,  or  the  time  of  a  complete  circuit 

9    C 

about  the  vertical  axis,  is ^ ,  so  that  when  the  rotational 

mgh 

velocity  is  high  the  precessional  velocity  is  very  slow. 

If  the  peg  of  the  top  is  rounded  and  the  surface  upon 
which  it  rests  rough,  so  that  there  is  no  slipping,  the 
conditions  are  different.  If  there  is 
a  vertical  axis  (Fig.  17)  through  a 
point  P  in  the  axis  about  which 
the  natural  precession  is  the  same 
as  the  forced  precession  due  to  the 
rolling  of  a  small  circle,  c,  of  the 
peg  on  the  surface,  and  if  the  top 
be  given  an  impulsive  velocity 
about  this  axis  such  that  the  gyro- 
scopic couple  exactly  balances  the 
gravitational    couple    about    the 

point  P,  then  the  motion  will  be  stable.  But  in  ordinary 
motion  with  nutations,  the  peg  would,  with  each  dip,  roll 
on  a  larger  circle,  thus  increasing  the  forced  precession,  and 
the  top  would  rise.  With  high  rotational  velocities  the  nat- 
ural precession  would  be  very  slow  while  the  forced  pre- 
cession would  be  rapid  and  consequently  the  top  would  rise 
rapidly.  Thus  while  a  top  with  a  peg  ending  in  a  mathe- 
matical point  cannot  rise  above  the  level  from  which  it  is 
let  go,  a  top  with  a  rounded  peg  on  a  rough  surface  will 
rise,  and  the  rise  is  at  the  expense  of 'the  rotational  energy. 
Brennan  has  applied  this  principle  of  forced  precession  to 
balancing  a  car  upon  a  single  rail  {v.  **The  Gyroscope"). 

30.   Drift  of  Rifled  Projectiles 

Fig.  18  shows  a  rifled  projectile  viewed  in  the  line  of 
flight  from  in  front,  its  long  axis  making  an  angle  t>  with 
the  path  of  the  centre  of  inertia,  0.  The  couple  due  to  the 
resistance  of  the  air  tending  to  restore  the  axis  to  the  line 
of  flight  increases  rapidly  up  to  a  certain  point  with  the 
angle  d^.  The  axis  and  the  path  at  first  coincide,  but  as  the 


74 


NATURAL   PHILOSOPHY 


trajectory  deviates  downward  from  its  original  direction, 
the  axis  fails  to  follow  it  and  a  couple  due  to  the  air  resist- 
ance strives  to  bring  it  in  line  again.-  If  from  any  point  the 
path  could  continue  as  a  straight  line,  the  axis  would  per- 
form a  regular  precession  with  its  nutations  about  this  line, 
as  indicated  in  Fig.  18.  It  would  preserve  a  constant  aver- 
age inclination,  ??,  to  the  path,  and  if  the  restoring  couple 

be  H  sin  t?  and  the  moment 
of  inertia  about  the  long 
axis,  C,  it  is  seen  from  Art. 
29    that    the    precessional 


DBIFT 


velocity  IS  — -^ and  the 

Co; 

time  for  a  complete  preces- 

2'kC(j3 


sion 


H 


But    the    line 


Fig.  18. 


about  which  the  long  axis 
strives  to  describe  a  regular 
cone  is  constantly  moving  downward.  When  the  point  of 
the  projectile  has  reached  its  lowest  point,  P\  the  line  of  the 
path  will  be  at  some  point,  0\  below  0,  and  if  the  path 
should  become  straight  from  this  point,  the  precessional 
cone  would  become  narrower.  It  is  by  such  an  action  that 
generally  the  long  axis  is  kept  close  to  the  line  of  flight.  As 
the  line  of  flight  moves  downward  the  angle  i?  is  always 
greater  in  the  upper  part  than  in  the  lower  part  of  the  pre- 
cessional cone.  This  has  an  important  frictional  result.  It 
will  be  noted  that  the  rotational  and  precessional  motions 
are  such  that  the  outer  (away  from  0)  surface  of  the  pro- 
jectile in  a  certain  sense  rolls  on  the  air  and  consequently 
there  is  little  air-friction  on  this  surface.  But  the  inner 
surface  (towards  0)  moves  against  the  air  with  both  its 
rotational  and  precessional  velocities.  The  air  is  thus  a 
smooth  surface  for  the  outer  surface  of  the  projectile, 
but  rough  for  its  inner  surface.  Since  the  precessional 
velocity  is  much  greater  in  the  upper  half  of  the  cone  than 
in  the  lower  half,  there  is  a  differential  effect,  the  friction 


DRIFT    OF    RIFLED   PROJECTILES  75 

on  the  inner  surface  in  the  upper  half  greatly  preponderat- 
ing. The  effect  is  the  same  as  if  the  projectile  were  laid  on  a 
partially  rough  surface  and  partly  slipped  and  partly 
rolled  parallel  to  its  long  axis.  If  when  viewed  from  behind 
the  projectile  is  rotating  to  the  left  —  positive  rotation  — 
the  precessional  motion  will  be  to  the  right,  and  the  "drift," 
or  the  horizontal  rolling  on  the  dense  underlying  cushion  of 
air  will  be  to  the  left.  The  axis  rolls  to  the  left  or  the  right 
according  to  the  rifling  but  always  remains  parallel  to  its 
original  vertical  plane. 

If  the  downward  velocity  of  the  path  is  equal  to,  or  not 
greater  than,  the  average  downward  velocity  of  the  axis  in 
its  precession,  they  will  keep  together,  nearly  meeting  at 
the  point  P'.  But  if  the  downward  velocity  of  the  path  is 
greater  than  this  limit,  so  that  when  P  reaches  P'  the  path  is 
below  this  point,  then  we  have  the  beginning  of  a  "tumble." 

The  precessional  velocity  is  proportional  to  H  sin  t> 
and  inversely  proportional  to  co.  There  is  consequently  a 
certain  limit,  readily  calculable,  beyond  which  the  axis 
cannot  keep  up  with  a  too  rapid  downward  curve  of  the 
path.  In  high  angle  (mortar)  firing,  at  the  vertex  of  the 
trajectory  the  curve  is  very  sharp  and  at  a  certain  limit, 
depending  upon  if,  w  and  r  where  r  is  the  angle  the  path 
makes  with  the  horizontal,  the  projectile  will  tumble. 
This  limit  will  be  reached  sooner  the  greater  the  value  of 
CO,  and  therefore  when  high  angles  are  used  a  great  amount 
of  rifling  is  not  desirable. 

There  have  been  many  misconceptions  as  to  rotary 
motion.  One  is  that  a  gyroscope  is  a  "gyrostat,"  or  device 
which  preserves  its  plane  of  rotation.  If,  by  using  a  mathe- 
matical fiction,  we  could  conceive  a  body  spinning  with  an 
infinite  velocity,  then  no  finite  couple  could  change  the 
direction  of  its  axis  and  it  would  be  a  "gyrostat."  But  the 
plane  of  any  finitely  spinning  gyroscope  is  readily  changed 
by  any  couple,  albeit  the  rate  of  change  diminishes  as  the 
rotation  increases.  Another  misconception  is  that  the 
rifling  of  projectiles  is  for  the  purpose  of  keeping  them 


76  NATURAL   PHILOSOPHY 

"end  on."  The  chief  advantage,  however,  of  rifling  is  to 
keep  the  projectile  in  the  gun  long  enough  to  have  the 
slow  burning  powder  develop  its  maximum  gas  pressure 
and  thus  launch  the  projectile  with  a  velocity  otherwise 
impossible.  A  spear,  any  long  fusiform  body,  even  an  un- 
tipped  arrow,  naturally  keeps  end  on.  They  are,  of  course, 
** stiff er,"  or  have  less  tendency  to  slew,  the  greater  the 
velocity,  but  any  slewing  tends  to  be  corrected  by  the  air 
couple.  For  high  angle  firing,  therefore,  if  it  were  possible 
to  delay  the  departure  of  the  projectile  until  the  full 
pressure  had  been  developed,  without  rifling,  a  long 
unrified  projectile  would  be  preferable  to  a  rifled  one, 
since  it  would  not  tumble,  it  would  not  wobble  about  the 
line  of  flight  as  a  rifled  projectile  does,  and  it  would  not 
"drift,"  thus  dispensing  with  an  otherwise  necessary 
correction. 

From  the  foregoing  it  is  evident  that  the  axis  of  a  rifled 
projectile  describes  a  path  such  as  that  shown  in  Fig.  19. 
The  main  curve  loops  downward  and  on  this  are  super- 
posed roulettes  (cycloids)  due  to  the  nutation — ripples 
as  it  were  on  the  principal  waves.  The 
reverberation  of  a  shell  as  it  passes,  in 
which   beats   are   distinctly  audible,  is 
OB/fT    ^^^  ^^  these  peculiar  vibrations  of  the 
^     axis  as  it  executes  the  major  loops. 

31.  Kepler's  Laws 

If  a  body  is  revolving  about  another, 
supposed  fixed,  and  we  consider  both 
Fig.  19.  bodies  simply  as  material  points,  then 

it  must  remain  in  a  fixed  plane  passing 
through  the  two  bodies,  and  its  moment  of  momentum 
about  the  central  point  must  remain  constant.  For  there  is 
no  couple  which  could  change  the  plane  or  moment  of 
momentum  of  the  revolving  body. 

If  it  describes  an  ellipse  about  the  attracting  point  and 
this  point  is  a  focus  of  the  ellipse,  the  attraction  must  vary 


KEPLER'S   LAWS  77 

inversely  as  the  square  of  the  distance.  For,  let  M  be  the 

mass  of  the  attracting  body  and  m  that  of  the  revolving 

body.  Writing  the  equation  of  the  ellipse, 

^    a  (1  -  e2) 

1  +ecosip  ^^^' 

where  a  is  the  major  semi-axis,  e  the  eccentricity,  and  <p 

the  angle  any  radius  makes  with  the  minimum  radius,  and 

designating  the  constant  moment  of  momentum,  m  r^  (p, 

by  N,  we  have  from  (1) 

Ne  sin  ^  .     ..  Ne 

w^  =  — ^^ ^  1  and  mr  =  —r. rr  cos  ^^. 

a  (1  -  e^y  a(l  —  e^)  ^^ 

The  radial  force  is  the  difference  between  the  centrifugal 

and  attractional  forces,  or  nir  =  mr^2  —  ^y^  where  /  is  the 

attractional  acceleration.     Hence 

47r2a^w  _  Mm 
r2T2     ~  l^r 

and 

The  attractional  force  is  therefore  inversely  as  the  square 
of  the  distance,  and  the  ratio  of  the  cubes  of  the  major 
semi-axes  to  the  squares  of  the  periods  is  the  same  for  all 
bodies  revolving  about  the  same  central  fixed  body. 

This  is  otherwise  evident  when  we  consider  that  the 
average  centrifugal  force  must  equal  the  average  attrac- 
tional force,  the  average  of  both  these  forces  being  that  at 
the  average  distance.  If  R  is  the  average  distance  and  n  the 
average  angular  velocity, 

2ir          .  72       47r2 
w  =^,and-^3  =-^  (3), 

where  M  is  the  mass  of  the  attracting  body.  This  shows  by 
(2)  that  the  average  distance  of  the  curve  from  a  focus  is  a. 
Otherwise,  as  the  sum  of  the  distances  of  any  point  from 
the  two  foci  is  always  2a,  the  average  distance  of  all  the 
points  from  one  of  the  foci  must,  by  symmetry,  be  a. 
The  above  results  are  Kepler's  Laws. 


a  (  -  e^)J  - 

-a(l 

-e2) 

mr^a  (1  — 

e') 

The  period. 

^    "      N 

2ira2  V 1  - 
1^ 

-  e^.m 

72 
n3 

47r2 

M 

(2) 

78  NATURAL   PHILOSOPHY 

Kepler's  laws,  however,  are  only  approximations,  since 
there  is,  of  course,  no  such  thing  in  Nature  as  a  mass  con- 
centrated into  a  point,  or  a  fixed  body.  When  one  of  the 
bodies  has  a  very  much  greater  mass  than  the  other  and 
the  distance  between  them  is  very  great,  as  is  the  case  for 
the  sun  and  a  planet,  these  laws  are  very  close  approxima- 
tions. Even  so,  each  body  describes  an  ellipse  which  has  the 
common  centre  of  inertia  of  the  two  bodies  for  a  focus. 
Considering  two  bodies  of  masses  M  and  m  to  be  centro- 
baric,  or  homogeneously  symmetrical  about  their  centres, 
the  ratio  of  their  accelerations  is  inversely  as  their  masses. 
If,  therefore  we  apply  to  both  bodies  the  acceleration 
of  one  of  the  bodies,  reversed  in  direction,  one  of  them 
will  be  brought  to  rest,  while  the  motion  of  the  other, 
relatively  to  it,  will  be  unchanged.  The  acceleration  of  one 
body  relatively  to  the  other  is  the  sum  of  their  accelera- 
tions, and  the  ratio  of  the  relative  acceleration  of  m  to 

its  actual  acceleration  is  — — — .  If  then  we  fix  M  and 

increase  its  mass  by  w,  the  relative  acceleration  of  m  will 
be  unchanged,  and  its  motion  relatively  to  M  will  be  the 
same  as  if  it  revolved  about  a  fixed  central  body  having  a 
mass  equal  to  the  sum  of  the  masses.  We  must  therefore 

write  instead  of    (3),   7^  =  71^-; ,   or  the  ratio  of  the 

R^      M  -\-  m 

cubes  of  the  major  semi-axes  to  the  squares  of  the  periods 

of  the  planets  is  proportional  to  the  sum  of  the  masses  of 

the  sun  and  each  planet,  and  not  simply  proportional  to 

the  mass  of  the  sun,  as  Kepler's  third  law  states. 

If  both  bodies  are  biaxial,  we  shall  see  later  that  this 

will  result  eventually  in  their  revolving  about  each  other 

in  a  plane  which  contains  both  equators  and  each  orbit  will 

be  a  perfect  circle  about  the  common  centre  of  inertia  for 

a  centre.   Both   before  and  after  the  merging  of  their 

equatorial  planes  Kepler's  third  law  would  not  be  strictly 

accurate. 


ATTRACTIONAL   HARMONICS  79 


32.   Attractional  Harmonics 

In  Art.  24  we  found  that  the  value  of  the  potential 
function  between  two  spheroids,  homogeneous  or  made  up 
of  homogeneous  confocal  shells,  was 

..^,„.,(_'),„@. 

where  F  (  )  is  a  series  with  ascending  powers  of  I  and  R 
respectively  in  the  numerators  and  denominators.  Let  us 
suppose  that  the  bodies  are  revolving  about  their  com- 
mon centre  of  inertia  in  circular  orbits  under  their 
mutual  gravitation.  In  investigating  the  motion  of  M 
relatively  to  M',  which  we  shall  consider  a  material 
point,  we  can  suppose  M'  to  be  fixed  and  to  have  a  mass 
M  +  M'.  If  7  be  the  declination  of  M\  then  the  couple 
tending  to  change  7,  or  the  rate  at  which  the  potential 

energy  is  used  up  about  the  axis  of  7  is  -j—,  and  this 

gravitational  couple  will  be  of  the  general  form 

■  dV  dT       ,^  dl         T.dl  ^    , 

G  =  -r-  =-  -a-j-  -\-  bl  -: cl2  _  4-  etc., 

07  07  ay  ay 

where  a,  6,  c,  etc.,  are  determinable  constants.  Since 
I  =  C  cos2  7  +  A  sin2  7  the  gravitational  couple  will  have 
the  general  form 

G  =  {a  ~  bl  -{-  cD  -  dP  +  etc.)  sin  7  cos  7  (1), 
where  a,b,  c,  are  other  constants.  It  will  be  noted  that  the 
couple  vanishes  when  R  passes  through  a  principal  axis  and 
therefore  altogether  when  the  body  is  uniaxial. 

We  shall  call  the  plane  containing  the  two  equal  axes 
the  equatorial  plane,  and  it  will  be  convenient  to  resolve 
the  gravitational  couple  into  a  couple  about  the  line  where 
the  equatorial  and  orbital  planes  intersect  —  the  nodal 
line  —  and  a  couple  about  an  axis  _L  to  this  in  the  equato- 
rial plane.  If  xp  is  the  precessional  velocity  of  the  C  axis 
about  the  orbital  pole,  we  may  call  the  first  axis  the  z?  axis 
and  the  second  axis  the  \p  sin  i?  axis. 


80 


NATURAL   PHILOSOPHY 


For  the  purposes  of  our  problem  it  is  indifferent  whether 
we  consider  M'  fixed  while  M  revolves  about  it,  or  M  fixed 

and  M'  revolving  in  the 
same  orbit.  Let  NAB  be 
the  orbit  of  the  attracting 
body,  E  its  pole,  O  the 
centre  of  M,  and  ON  the 
nodal  line.  We  shall  sup- 
pose the  orbit  to  be  a 
perfect  circle.  Let  the  at- 
tracting body  be  at  A  and 
moving  with  constant  an- 
gular velocity,  a.  Taking 
our  measures  from  N,  since 
xj/  is  small  compared  with 
a,  we  can  for  a  short  time  consider  the  angle  NO  A  as 
sensibly  equal  to  at.  y  =  POA,  and  the  gravitational  couple 
is  in  this  plane.  Let  the  angle  between  the  planes  AOP  and 
BOP  be  (p.  Then  the  4  component  is  G  cos  (p  and  the  xp  sin  t^ 
component  is  G  sin  ^.  By  spherical  trigonometry, 

cos  ^  .    .  ctn  at 

(p  =  and  sm  <p  =     ,  • 

'  ~    '         '  Vcos2  d^  +  ctn^at 

Vl  —  sin2  d^  sin2  at. 


cos 


Vcos2  ^  -\-  ctn^at 
cos  7  =  sin  ??  sin  at  and  sin  y 

Since 


sm  at 


f-. 


sin2  ??  sin2  at 


cos2  i}  +  ctn^at 
the  t?  component  of  the  term  kl"  sin  y  cos  y  in  (1)  be- 
comes kl^  sin  ^  cos  t?  sin2  a^,  and  the  \p  sin  ?>  component 
becomes  kl"  sin  t?  sin  at  cos  a/. 

Since  the  nutation  is  small  compared  with  ??,  we  may 
consider  t?  as  sensibly  constant,  and  since 

I  =  A  +  {C  -  A)  sin2  ??  sin2  a^ 
the  total  4  component  can  be  thrown  into  the  general  form 

AS^  =   —bi   sin2   at  +  62    sin*    at  —  63    sin^   at 

^  b„  sin2«  at  -f  etc.  (2). 

Likewise  the  xp  sin  ??  component  can  be  thrown  into  the 
general  form 


\ 


ATTRACTIONAL  HARMONICS  81 

A^p  sin  d^  =  ct  sin  at  eos  at  —  C2  sin^  at  cos  at  + 
C3  sin^  a^  cos  at  ...  .    =±=  Cn  sin^^*-^  ai  cos  at  +  etc.     (3). 
sin2n  at  can  be  written  as  the  limited  series, 
sin2»  at  =  k  —  ki  cos  2  at  -{•  kz  cos  4  ai  —  ^3  cos  6  at .  .  .  . 
=t  kn  cos  2m  a^  (4). 

For  cos  2  a/  =  1  —  2  sin2  a^ 

cos  4  a^  =  1  —  8  sin2  at  -{■  S  sin^  a^ 
and  so  on. 
Conversely,  cos  2n  at  can  be  written  as  the  limited  series, 

cos  2n  at  =  m  —  mi  sin2  at  +  m2  sin^  at  —  mz  sin*?  at 

=t  mn  sin^**  at.  (5). 

Hence  the  t?  couple  has  the  general  form 

i4t?  =  6—61  cos  2  a^  +  ^2  cos  4  ai  —  63  cos  6  a^ 

=t  6„  cos  2n  a^  (6). 

By  differentiating  (4)  we  have  the  series 
sin2«-i  ^^  (.Qg  ^^  ^  ^j  gj[j;^  2  at  —  ci  sin  4  af  +  ^3  sin  6  a^  .  .  .  . 
=±=Cn  sin  2m  a^  (7), 

and  we  can  throw  the  yj/  sin  t?  couple  into  the  general  form 
of  (7). 

Combining  any  two  terms  of  the  series  (6)  and  (7)  having 
the  same  period,  it  is  evident  that  we  have  an  elliptic  har- 
monic motion,  the  axes  of  the  ellipse  being  determinable 

in  each  case.  The  fundamental  period  is  -,  a  being  the 

a 

angular  velocity  of  the  attracting  body,  and  this  is  the 

period  of  the  first  ellipse  due  to  the  two  couples,  —  hi  cos  2  at 

and  c\  sin  2  at.  The  following  ellipses  represent  the  higher 

harmonics  of  this  fundamental  period,  viz., 

TT          TT          TT         TT 
—      ^__       PTP 

2a'  2>a'  4a'  5a' 

Fig.  21  represents  the  first  four  ellipses  viewed  from 
outside  the  orbit.  The  long  arrow  shows  the  direction  of 
motion  of  the  attracting  body.  The  first  ellipse  is  vertical, 
the  second  horizontal  and  so  on  alternately,  the  motion  in 
the  vertical  ellipses  being  always  to  the  left,  while  that  in 
the  horizontal  ellipses  is  to  the  right. 

Taking  the  case  of  the  earth  and  the  moon,  since  the 


82  NATURAL   PHILOSOPHY 

distance  between  these  bodies  is  great,  the  gravitational 
series  (1)  decreases  very  rapidly  and  the  first  ellipse  is  the 
only  one  which  is  appreciable.  Let  us  investigate  this 
ellipse.  Taking  the  first  term  of 

(M  +  Ml)  F  (|\ 

we  readily  find  that  At?  =  -X  sin  t?  cos  t?  sin2  at  (8) 
and  i4i//  sin  t?  =  K  sin  d^  sin  at  cos  at  (9), 


0 


o 


V         Tl 


The  elliptic  motion  due  to  these  two  couples  can  be  repre- 
sented by  a  material  point,  or  particle,  moving  harmoni- 
cally in  a  vertical  ellipse,  to  the  left,  with  constant  angular 

velocity  2a.  The  major  axis  of  the  ellipse  is  ^rp-  sin  t>, 

IX- 

and  the  minor  axis  is  tt?^  sin  t?  cos  t?.  The  vertical  veloc- 
zGco 

ity  in  the  ellipse  will  be  t?  =  —  tt^  sin  t?  sin  2  at  and 

the  horizontal  velocity,  xp  sin  t>  =  tttt-  sin  t?  cos  t?  cos  2  a^ 
If  now  we  suppose  the  ellipse  to  move  bodily  to  the  left 

zx- 

with  a  horizontal  angular  velocity  —  7^7=^  sin  t>  cos  z?,  the 
total  horizontal  velocity  of  the  particle  will  be  yp  sin  t?  = 


ATTRACTIONAL   HARMONICS  83 

K  K 

777T-  sin  ^  cos  I?  (cos  2  at  —  1)  =  —  7;-  sin  t?  cos  ^  sin2  at. 

Substituting  the  extremity  of  the  axis  of  the  earth  for  the 
particle,  the  horizontal  angular  velocity  xp  sin  t>  will  gyro- 
scopically  cause  a  vertical  angular  acceleration, 
K  sin  ^  cos  t?  sin2  at, 

and  the  vertical  velocity,  t?  =  —  ^r^  sin  t?  sin  2  a/,  will  cause 

a   horizontal   angular   acceleration  —  ^^  sin  t?  sin  2  at. 

But  these  gyroscopic  couples  are  exactly  equal  and  op- 
posite to  the  gravitational  couples  (8)  and  (9).  We 
have  seen  that  gyroscopic  couples  are  purely  internal 
forces,  representing  merely  the  moments  of  the  centrifugal 
forces,  which  are  forces  of  inertia.  The  gravitational 
couples  are  the  applied  forces  and  the  gyroscopic  couples 
are  the  forces  of  inertia,  due  to  the  motion.  These  forces  are 
exactly  balanced  and  therefore  the  axis  of  the  earth  moves 
freely  (without  constraint)  in  the  first  ellipse.  The  axis 
of  the  earth  executes  a  harmonic  motion  in  the  first  ellipse 
with  constant  angular  velocity,  2a,  in  a  counter  clockwise 
direction,  while  the  ellipse  itself  performs  a  constant 
horizontal  retrograde  precession  about  the  pole  of  the 

moon's  orbit  with  angular  velocity,  ^o  =  —  97^  cos  t?o> 

where  1^0  is  the  constant  inclination  of  the  centre  of  the 
ellipse. 

In  Equa.  (6)  there  is  a  single  unpaired  term,  b.  This  is 

equal  to  ^0  sin  ??o  =  —  — r-  sin  ^0  cos  t?o-  The  motion  of  the 
zGco 

mean  position  of  the  axis,  t?o,  thus  produces  a  gyroscopic 

couple,  Cwi/'o  sin  t?o,  which  exactly  balances  the  gravita- 

tional  couple  for  this  inclination,  viz.,  —  y  sin  t?o  cos  ^oy 

with  a  resulting  constant  and  smooth  precession  of  this 
mean  position,  viz.,  the  centre  of  the  harmonic  ellipse. 
It  will  be  noted  that  in  deriving  Equa.  (6)  every  term 


84  NATURAL   PHILOSOPHY 

gave  a  harmonic  not  only  of  the  order  of  the  term  but  also 
of  all  the  lower  harmonics,  together  with  a  constant 
(zero  harmonic)  which  is  a  part  of  6,  the  couple  pro- 
ducing the  constant  retrograde  precession.  The  constant 
precession  is  therefore  represented  by  a  series  made  up  of 

alternating  plus  and  minus  terms,  and  the  value  derived 

j^ 

from  the  first  ellipse  alone,  viz.,  xf/o  =  —  ^r^-  cos  t?o,  is 

zGco 

slightly  greater  than  the  actual  value.  The  actual  motion  is 

the  following:  The  centre  of  the  first  ellipse  executes  a 

constant  retrograde  horizontal  precession.  A  point  in  this 

ellipse  moves  with  a  constant  angular  velocity,  2a,  in  a 

positive  direction  (to  the  left).  About  this  point  another 

point  describes  the  second  ellipse  with  an  angular  velocity 

4a  in  a  negative  direction.  About  the  last  point  another 

point  describes  the  third  ellipse  with  an  angular  velocity 

6a,  and  so  on,  while  the  axis  of  the  earth  moves  in  the  last 

ellipse  of  all. 

Whenever  the  couple  ceases,  the  axis  comes  instantly  to 

rest,  and  starting  from  any  position  it  immediately  falls 

into  motion  in  that  part  of  the  harmonic  ellipse  necessary 

to  bring  it  to  rest  at  the  node.  Considering  only  the  first 

ellipse,  the  inclination  of  the  axis  to  the  pole  of  the  orbit 

is  always  greatest  at  the  nodes  and  least  at  quadratures. 

That  is,  the  axis  is  always  at  the  bottom  of  the  ellipse  at 

the  nodes.  Where  two  biaxial  bodies  revolve  about  each 

other  at  a  distance  which  is  not  an  excessive  multiple  of 

their   diameters  —  and  there  are  such  instances  among 

heavenly  bodies  —  not  only  the  fundamental  period  but  a 

number  of  the  higher  harmonics  would  be  appreciable, 

constituting  a  veritable  "Music  of  the  Spheres." 

33.   Simplification  of  Motion 

In  investigating  the  precession  of  a  planet  we  saw  that 
the  gravitational  couple  tending  to  bring  the  equatorial 
plane  into  the  line  joining  the  centres  of  inertia  of  the  two 
bodies  was 


SIMPLIFICATION   OF   MOTION  85 

K  sm  7  cos  7  =  ^3 (C  —  A)  sm  7,  cos  7, 

where  7  is  the  declination  of  the  attracting  body.  But 
the  action  is  mutual  and  an  equal  couple  strives  to 
bring  the  attracting  body  into  the  plane  of  the  equator. 
This  couple  causes  the  plane  of  the  orbit  to  precess  about 
the  pole  of  the  planet  just 
as  the  attractional  couple  y/^    /    a 

of  the  satellite  causes  the 
polar  axis  of  the  planet  to 
precess  about  the  pole  of 
its  orbit.  Let  us  suppose 
a  satellite,  Fig.  22,  to  re-  p^^^  22 

volve  about  a  spheroidal 
planet  at  a  constant  distance  from  0.  If  the  planet  were 
a  sphere,  the  satellite  would  describe  a  circular  orbit 
NAN'.  The  component  of  our  couple  in  the  direction 
of  the  path  of  the  satellite  is  K  sin  7  cos  7  cos  r,  where 
T  is  the  angle  the  path  makes  with  a  meridian.  The 
velocity,  instead  of  being  constant,  will  therefore  be 
retarded  from  N  to  A  and  accelerated  from  A  to  N\ 
but  the  velocity  in  the  equator  will  always  be  the  same, 
viz.,  the  velocity  for  the  circular  orbit.  It  is  evident,  there- 
fore, that  a  satellite  cannot  describe  a  circular  orbit  about 
a  biaxial  body,  unless  it  keeps  in  the  equatorial  plane.  It 
executes  a  path  NBC  wholly  within  the  circular  orbit, 
and  making  the  same  angle  with  the  equatorial  plane  at 
N  and  C  as  the  uninfluenced  path  NAN\  The  node  C 
occurs  before  the  node  N'  and  the  effect  of  the  couple 
is  to  make  the  nodes  regress,  while  the  average  inclination 
of  the  path  to  the  plane  of  the  equator  and  the  average 
velocity  remain  constant.  The  inclination  of  the  plane  of 
the  path  to  the  plane  of  the  equator  is  least  at  the  summit, 
B,  and  greatest  at  the  nodes,  where  it  is  the  constant  in- 
clination of  the  uninfluenced  path.  The  motion  of  the  orbit 
is  the  same  as  that  of  a  solid  ring,  into  which  we  may  sup- 
pose the  mass  of  the  satellite  to  be  uniformly  distributed^ 


86  NATURAL   PHILOSOPHY 

rotating  with  the  same  angular  velocity.  A  point  on  this 
ring  gives  the  position  of  the  satellite  at  any  time.  The 
motion  of  such  a  ring  is  obviously  a  precession  of  its  axis 
about  the  pole  of  the  planet,  accompanied  by  nutations, 
precisely  as  in  the  case  of  a  top. 

The  above  is  on  the  supposition  that  the  satellite  main- 
tains a  constant  distance  from  the  planet.  Actually  such  a 
condition  is  only  possible  when  the  orbit  coincides  with  the 
equatorial  plane. 

Actually  the  orbit  not  only  precesses,  but  gradually 
loses  its  inclination  until  it  finally  coalesces  with  the  equa- 
torial plane,  in  somewhat  the  same  way  as  a  plate  spinning 
on  its  edge  on  a  table  eventually  coincides  with  the  table. 
A  biaxial  body  eventually  brings  any  revolving  body 
permanently  into  the  plane  of  its  equator.  Rigorous  proofs 
of  this  have  been  given  by  Laplace  and  Tisserand.  An  in- 
formal explanation  may  be  found  in  "Popular  Astronomy," 
Sept.  1915.*  Having  got  our  satellite  into  the  equatorial 
plane,  let  us  see  what  happens  next.  A  body  launched  in 
the  equatorial  plane  will,  in  general,  describe  an  orbit 
which  is  nearly  an  ellipse,  although  not  exactly  an  ellipse, 
provided  the  planet  is  biaxial.  Taking  first  the  case  of  a 
spherical  planet  of  mass,  M,  the  orbit  will  be  an  ellipse 

and  by   Art.    32   it  is  readily  seen   that   —  =  -rrz , 

where  ri  is  the  least  distance  of  the  orbit  from  the  planet, 
n  the  greatest  distance,  and  A^  =  r2^.  if  now  we  suppose 
the  planet  to  be  slightly  flattened,  the  attraction  at  all 
points  in  the  equatorial  plane  will  be  increased,  so  that 
starting  from  n  the  corresponding  maximum  distance 
r2^  will  be  shorter  than  rz.  The  path  will  be  nearly  an 
ellipse  corresponding  to  a  slightly  greater  mass  at  the 
focus.  Likewise  starting  from  r2,  the  corresponding 
minimum  distance  n^  will  be  shorter  than  n. 

1    _  2  (M  +  dM)       1  1        2  (M  +  dM)       1 

rii  A^2  r2'  n'  "  m  n* 

*  Some  problems  in  Gravitational  Astronomy. — The  Author. 


SIMPLIFICATION    OF   MOTION  87 

Whence  r  = and  —  >  —  (1). 

That  is,  owing  to  the  flattening  of  the  planet,  the  satel- 
lite describes  an  approximate  ellipse  with  a  major  axis 
which  is  slightly  less  than  that  of  the  original  ellipse  and 
the  maximum  distance  from  the  focus  is  decreased  by  a 
greater  amount  than  the  minimum  distance.  Calling  the 
eccentricity  of  the  original  ellipse,  e,  and  that  of  the  new 

1  —  ^1        \  —  e 

approximate  ellipse,  e^,  we  have  from  (1)  :; — ■ — -  >  -. — ; — , 

1+^1        \  -\-  e 

ox  e  >  el,  or  the  new  approximate  ellipse  is  less  eccentric 
than  the  original  one.  It  will  be  seen  that  in  the  inward 
journey  the  new  path  is  within  the  original  ellipse,  while  on 
the  outward  journey  it  is  without.  Thus  the  inward  half  of 
the  new  path  is  more  eccentric  than  the  old  path,  while  the 
outward  half  is  less  eccentric,  but  on  the  whole  the  new 
path  is  less  eccentric.  It  is  further  evident  that  owing  to 
the  greater  eccentricity  of  the  inward  half,  the  minimum 
radius  will  be  slightly  ahead  of  the  old  one,  while  owing  to 
the  lesser  eccentricity  of  the  outward  half 
the  maximum  radius  will  be  behind  the 
old  one. 

In  other  words,  the  major  axis  pro- 
gresses at  minimum  distance  and  re- 
gresses at  maximum  distance,  but  the 
former  exceeds  the  latter,  so  that  on  the 
whole  the  major  axis  progresses,  or  moves 
in  the  direction  of  the  motion  with  each 
revolution.  In  Fig.  23,  the  full  line  rep- 
resents the  original  ellipse  and  the  dotted 
line  the  transformed  path.  We  see  then  that  a  satellite 
revolving  in  the  equatorial  plane  of  a  biaxial  planet:  1. 
Alternately  increases  and  decreases  its  eccentricity,  but  on 
the  whole  progressively  decreases  it,  until  it  revolves  in 
a  perfect  circle.  2.  The  approximate  major  axis,  by  alter- 
nate progressions  and  regressions,  on  the  the  whole  pro- 
gresses. 3.  The  semi-major  axis,  or  mean  distance,  pro- 


88  NATURAL   PHILOSOPHY 

gressively  decreases  until  the  final  circular  orbit  is  attained. 
Better,  however,  than  any  theoretical  proof  is  the  direct 
experimental  proof  which  meets  our  eyes  at  many  points 
of  the  heavens.  All  the  nearer  satellites  revolve  nearly 
in  their  equatorial  planes  in  almost  perfect  circles,  and  they 
would  perform  these  motions  exactly  if  it  were  not  for  the 
disturbing  action  of  the  sun.  There  can  be  no  more  beauti- 
ful experiment  than  the  following:  Suppose  some  power 
able  to  hurl  masses  of  matter  at  some  planet  isolated  in 
space.  The  planet  would  catch  them,  and  winding  them 
about  itself  would  gradually  bring  them  all  into  its 
equatorial  plane,  moving  in  nearly  perfect  circles.  A  single 
satellite  would  describe  an  exact  circle. 

We  shall  see  directly  that  the  disturbing  action  of  the 
sun  causes  the  orbits  of  satellites  to  assume  a  compromise 
position  between  the  equatorial  and  orbital  planes  of  the 
planet.  The  plane  about  the  axis  of  which  an  orbit  per- 
forms its  precessions  is  called  the  fundamental  plane  of 
the  orbit.  It  is  not  necessary  that  the  influencing  body 
should  be  within  the  orbit,  for  a  distant  body  can  like- 
wise produce  a  precession  of  the  orbit.  The  sun  causes  the 
moon's  orbit  to  precess  exactly  as  the  earth's  equatorial 
protuberance  does,  and  it  happens  that  the  sun's  influence 
is  considerable,  due  to  his  great  mass,  while  the  earth's 
influence  is  slight.  We  may  represent  such  a  precession  by 
a  vector  perpendicular  to  its  fundamental  plane,  having  a 
length  equal  to  the  precessional  velocity,  and  in  the  case  of 
several  influencing  bodies  we  can  compound  the  effect  by 
compounding  the  vectors.  Thus  the  precession  of  the 
moon's  orbit  due  to  the  sun  has  the  ecliptic  for  its  funda- 
mental plane,  while  the  precession  due  to  the  earth  has  the 
earth's  equatorial  plane  for  its  fundamental  plane.  The 
resultant  precessional  axis  lies  in  a  plane  containing  the 
axis  of  the  ecliptic  and  the  earth's  polar  axis,  and  inclined 
to  the  former  about  12'.  The  inclination  of  the  moon's 
orbit  to  this  resultant  axis  is  about  84°  40'.  Hence  as  the 
moon's  orbit  rotates  about  this  resultant  axis,   its  in- 


SIMPLIFICATION    OF   MOTION  89 

clination  to  the  ecliptic  varies  from  a  maximum  of  5°  20'  to 
a  minimum  of  4°  56'. 

This  effect  of  the  equatorial  protuberance  of  a  planet  in 
bringing  a  satellite  into  its  plane  and  then  destroying  its 
eccentricity,  is  very  strong  when  the  satellite  is  near  the 
planet.  It  is  strikingly  shown  in  the  case  of  the  satellites 
of  Mars  and  of  all  the  nearer  satellites  of  our  system. 
The  equatorial  planes  of  the  planets  are,  of  course,  con- 
stantly shifting,  due  to  planetary  precession,  but  they 
carry  their  nearer  satellities  with  them  practically  the  same 
as  if  their  orbits  were  rigidly  attached. 

These  gravitational  effects  all  exemplify  a  general 
principle  in  Nature  which  we  may  call  the  Simplification  of 
Motion.  There  is  everywhere  a  tendency  to  reduce  com- 
plicated and  irregular  forms  of  motion  to  simpler  and  more 
regular  forms.  By  the  development  of  gyroscopic  couples, 
two  or  more  rotations  tend  to  fuse  into  a  single  rotation. 
This  tendency  may  result  only  in  an  oscillation  about  the 
position  of  fusion  (equilibrium)  but  frictional  forces 
eventually  effect  the  fusion.  The  motion  of  a  triaxial  body 
with  its  instantaneous  axis  in  the  separating  polhode  is  an 
example  of  the  simplification  of  motion.  Tidal  forces  tend 
to  equalize  rotational  and  revolutional  motions  and  even- 
tually do  equalize  them  —  this  being  the  simplest  form  of 
such  a  double  motion.  We  shall  see  directly  that  all  ro- 
tational planes  tend  to  coalesce  with  revolutional  planes. 

The  first,  second,  and  third  satellites  of  Jupiter  are  an 
example  of  the  harmonizing  of  motions.  Considering  a 
revolution  as  a  vibration,  and  circular  orbits  are  composed 
of  two  simple  harmonic  motions  perpendicular  to  each 

other,  the  frequencies  of  these  vibrations  are  ^,  ^,  and  -=-, 

i  1     i  2  ^3 

where  T  is  the  period.  By  their  mutual  interactions,  they 
have  been  able  to  bring  their  frequencies  into  a  simple 
harmonic  relation.   The  frequencies  are  very  nearly  as 

1,  ^>  J-  The  five  inner  satellites  of  Saturn  have  frequencies 


90  NATURAL   PHILOSOPHY 

nearly    as    ^^1,    1,    1    .  .  .  1  .  .  .  ^.    A    vibrating 

body  not  only  tends  to  set  up  harmonic  vibrations  in  other 
bodies,  but  when  that  is  impossible  and  the  shape  is 
changeable,  actually  tends  to  shake  them  into  forms  capa- 
ble of  such  harmonics.  A  rigid  body  can  only  respond  to 
certain  fixed  frequencies,  but  an  elastic  body  may  adjust 
itself  to  the  proper  frequencies. 

The  mutual  tendency  of  the  orbits  of  our  system  is  to 
coalesce  into  a  single  plane,  and,  given  time  enough,  they 
will  eventually  coalesce  into  the  Invariable  Plane  of  the 
system.  And  nowhere  is  there  an  opposite  tendency.  Simple 
and  regular  motions  never  degenerate  into  complex  and 
irregular  forms,  for  the  simpler  the  motion  the  more  stable 
does  it  become,  and  all  irregular  and  complex  forms  are 
essentially  unstable. 

34.   Effect  of  Moon's  Orbital  Precession  on  Earth's  Axis 

Owing  chiefly  to  the  sun's  action,  the  moon's  orbit  per- 
forms  a   complete   precession,    with   the   ecliptic   as   its 

fundamental  plane,  in   about 

/^  ^s.  18^  years.  This  in  turn  exerts 

\        an    influence   on    the    earth's 
^  \     axis  which  we  shall  now  ex- 

amine. Let  A,  Fig.  24,  be  the 
position  of  the  earth's  axis, 
C  the  pole  of  the  ecliptic,  d^ 
the  angle  CA,  B  the  pole  of 
the  moon's  orbit,  and  CB  the 
constant  angle,  a.  Actually  d^ 
is  about  23°  27'  and  a  is  nearly 
5°.  It  is  evident  that  we  can 
effect  the  steady  retrograde 
precession  caused  by  a  revolving  body,  by  fixing  half  its 
mass  at  the  pole  of  its  orbit  and  supposing  it  to  exert  a 
repellent  instead  of  an  attractive  action.  We  have  then  to 
consider  the  action  of   a  body  having   half   the  moon's 


MOON'S   ORBITAL   PRECESSION  91 

mass  and  moving  in  a  retrograde  direction  in  the  small 
circle  with  constant  angular  velocity,  —b.  Let  the  angle 
BA  be  c.  Then  the  gravitational  couple  is  —K  sin  c  cos  c. 
If  the  angle  CAB  =  A,  the  &  and  rp  sin  t?  couples  are 

—  K  sine  cos  c  cos  A  and  K  sin  c  cos  c  sin  A . 

Since  yp,  the  precession  of  the  earth's  axis  about  C,  is 

small  compared  with  6,  the  angle  ACB  =  C,  measured  from 

a  position  of  conjunction,  will  for  a  short  time  be  sensibly 

-     sin  A       sin  a  ...        - 
equal  to  —  bt.  — — 7^  =  ~. —  (1)  and  cos  c  =  cos  a  cos  t}  + 
^  sm  C      sm  c 

sin  a  sin  ^  cos  C  (2). 

Hence  the  

t^  couple  is  —  K  cos  c  V  sin2  c  —  sin2  a  sin2  C. 

sin2  c  —  sin2  a  sin2  C  =  (cos  a  sin  ^  —  sin  a  cos  t?  cos  C)2^ 

and  the  t?  couple  is 

—  K"  cos  c  (cos  a  sin  t?  —  sin  a  cos  i}  cos  C) . 

Substituting  the  value  of  cos  c  from  (2)  this  becomes 

A^  =  —K  cos2  a  sin  t?  cos  t?  +  i^T  sin  a  cos  a  cos  2  t?  cos  C  + 

K  sin2  a  sin  ??  cos  t?  cos2  C  (3).  Likewise 

Axp  sin  i>  =  K"  sin  c  cos  c  sin  A  =  K  sin  a  cos  a  cos  z?  sin  C  + 

K'  sin2  a  sin  z?  sin  C  cos  C  (4) . 

Our  equations  of  motion  therefore  are 

K  sin2  a  sin  z?  cos  t>  cos2  bt  -]-  K  sin  a  cos  a  cos  2  t?  cos  6i  — 

K  cos2  a  sin  t?  cos  t?  -  Ccoxp  sin  t?  +  i4i/'2  sin  ??  cos  t>  =  At?.  (5) . 

—  K"  sin2  a  sin  t?  sin  6^  cos  bt  —  K  sin  a  cos  a  cos  t?  sin  6^  + 

Cco^  -  Ai^  cos  M  =  A)A  sin  t?  (6). 

[We  have  dropped  the  angles  A  and  C  and  these  symbols 
now  resume  their  usual  significance.] 

4  and  \p  are  so  small  that  we  can  neglect  second  powers  and 
T?  is  sensibly  constant.  The  motion  is  so  small  that  we  can 
use  X  in  place  of  xp  sin  t?  and  y  in  place  of  ??.  In  other  words 
we  can  use  rectangular  in  place  of  spherical  co-ordinates. 
Hence  our  equations  of  motion  become, 
K  sin2  a  sin  t>  cos  t?  cos2  bt  -{-  K  sin  a  cos  a  cos2  t?  cos  6^  — 

K  cos2  a  sin  t?  cos  t?  —  Co3X  =  Ay  (7) 
and  —  K  sin2  a  sin  t?  sin  6/  cos  bt  —  K  sin  a  cos  a  cos  i>  sin  6/  + 
Cw>'  =  Ax  (8). 


92  NATURAL   PHILOSOPHY 

Integrating    (8),    Ccoy  —  A    {x  —  Xo)  = 

r^   •  o       •     «  sin2  bt 
K  sm^  a  sm  t? 


K  sin  a  cos  a  cos  ?> 


26 
(  cos  bt  -  1) 


6 

Since  x  is  small  compared  with  co,  we  can  write  this 

^           r^    •  o       •     o  sin2  bt    , 
C(i}y  =  A.  sin2  a  sm  t?  — yr h 

T^    '                         0  (1  ~  cos  bt)  .^. 

A  sm  a  cos  a  cos  t>  ^ r (9). 


Integrating  (7), 

^0 


^           rx    •   ->       •     o          Q  I  '    .     (sin6/cos60\    , 
Ccoic  =  K  sm2  a  sm  I?  cos  t?  (7^  H ^T ^  )  + 


K  sin  a  cos  a  cos  2  t>  — r —  —  K  cos2  a  sin  ^  cos  ??.  /      (10). 

Plotting  the  curve  from  these  equations  we  find  that  it 
has  the  shape  given  in  Fig.  24. 

Starting  with  the  origin  at  the  time  when  the  poles 
are  on  the  same  celestial  meridian  (conjunction),  the 
inclination  of  the  earth's  axis  to  the  pole  of  the  ecliptic  is 
here  a  minimum.  After  this  it  increases  until  a  maximum  is 
reached  with  bt  =  x.  It  then  regains  its  former  minimum 
when  bt  =  It  and  the  two  poles  are  again  in  conjunction. 
The  path  of  the  earth's  axis  is  thus  an  unsymmetrical 
wavy  curve.  There  are  about  1400  such  complete  waves  in 
every  complete  precessional  circle  of  the  earth's  axis  about 
the  pole  of  the  ecliptic.  The  variation  of  t?,  or  the  depth  of 
the  curve  is  about  9''. 

35.   Effect  of    the   Moon*s   Orbital    Precession   on  her 

Own  Axis 

This  action  upon  the  earth's  axis,  due  to  the  shifting  of 
the  moon's  orbit,  is  purely  reciprocal. 

The  earth's  attraction  upon  the  moon's  equatorial  pro- 
tuberance causes  the  moon's  axis  to  precess  (retrograde) 
about  the  pole  of  her  own  orbit.  As  far  as  the  precessional 
effect  is  concerned,  it  is  a  matter  of  indifference  whether 


EFFECT   ON   HER   OWN    AXIS  93 

the  moon  revolves  about  the  earth  or  the  earth  revolves 
about  the  moon  in  the  moon's  orbit.  We  can  cause  the  same 
precessional  effect  upon  the  moon's  axis  by  supposing 
half  the  earth's  mass  to  be  at  the  pole  of  the  moon's  orbit 
exerting  a  repulsional  instead  of  an  attractional  action. 
If  then  in  Fig.  24  we  suppose  A  to  be  the  pole  of  the  moon's 
orbit  moving  with  a  constant  retrograde  precession,  —  6, 
and  half  the  earth's  mass  to  be  at  this  pole,  while  the 
moon's  axis  is  at  B,  the  problem,  though  reversed,  is 
exactly  similar  to  the  previous  one.  The  action  of  the 
earth  will  be  to  cause  the  moon's  axis  to  precess  in  the 
small  circle,  B,  with  nutations,  forming  a  wavy  curve 
precisely  as  in  the  other  problem.  Calling  now  the  angle 
CA,  ??,  and  the  angle  CA,  a,  the  equations  of  motion  are 

K  sin2  a  sin  t?  cos  t?  cos2  {y^/  —  hi)  -f- 
K  sin  a  cos  a  cos  2  ^  cos  (i/'  —  bt)  —  K  cos2  a  sin  t^  cos  x?  — 
Ccot/'sint?  =  A^  (11)  and 
K  sin2  a  sin  ??  sin  (^  —  bt)  cos  (yj/  —  bt)  -f 
K  sin  a  cos  a  cos  ^  sin  (^  —  bt)  +  Ccor?  =  Axf'  sin  r>    (12), 
where  K  and  rp  now  refer  to  the  moon.  The  angle,  xf/  —  bt^ 
is  the  difference  between  the  precession  of  the  moon's  axis 
and  the  precession  of  the  pole  of  her  orbit. 

Calling  ^  —  bt,a,  the  equations  of  motion  can  be  written 
D  cos2  a  +  Ecosa  -  F  -  Cco^  sin  t?  =  At?  (13). 
G  sin  a  cos  a  +  ^  sin  a  +  Ccoz?  =  A4^  sin  t?  (14), 
where  D,  E,  F,  etc.,  are  determined  constants.  The  natural 
independent  precessions  of  the  two  poles  we  are  consider- 
ing, viz.,  xj/  the  precession  of  the  moon's  axis  and  bt  the 
precession  of  the  pole  of  her  orbit,  are  not  the  same. 
However,  they  are  forced  into  coincidence  by  a  peculiar 
action  which  we  shall  now  discover.  The  natural  pre- 
cessions being  different,  one  pole  will  eventually  overtake 
the  other  and  at  some  time  a  will  be  momentarily  zero, 
and  at  some  other  time  momentarily  tt  —  conjunction  or 
opposition.  From  Fig.  24,  we  see  that  in  these  positions 
!?  is  momentarily  zero. 

Let  us  suppose  that  a  has  become  tt,  the  two  poles 


94  NATURAL   PHILOSOPHY 

being  at  their  maximum  distance  apart,  with  C  the  pole  of 
the  ecliptic  between  them.  From  (14),  the  precessional 
acceleration,  \j/  sin  t?,  is  here  zero,  and  the  precessional 
velocity  is  momentarily  constant.  But  ^  and  h  not  being 
equal,  the  moon's  axis  will  directly  either  get  ahead  of  or 
lag  behind  the  orbital  pole.  If  the  angle  a  becomes  negative, 
from  (14)  the  precessional  acceleration  becomes  negative 
and  a  couple  is  brought  into  play  tending  to  bring  the 
moon's  axis  back  into  coincidence  (opposition)  with  the 
orbital  pole.  If  the  moon's  axis  gets  ahead  and  the  angle  a 
becomes  positive,  from  (14)  a  positive  couple  arises  tending 
to  turn  the  moon's  axis  back  into  coincidence  with  the 
other  pole. 

There  is  a  limit  beyond  which  this  regulatory  couple 
could  not  overcome  the  difference  between  the  natural 
velocities,  but  in  the  case  of  the  moon,  her  mass  being 
slight,  the  couple  is  well  within  this  limit. 

When  a  is  zero,  if  the  moon's  axis  gains  from  this 
position,  a  becomes  negative  and  a  negative  couple  arises 
which  tends  to  increase  the  gain  still  further,  while  if  the 
moon's  axis  lags,  a  becomes  positive  and  a  positive  couple 
arises  which  tends  to  set  it  still  further  back.  Hence, 
when  the  two  poles  are  in  conjunction  they  exert  a  mutually 
repellant  action  and  are  in  unstable  equilibrium,  while 
when  they  are  in  opposition  they  are  in  stable  equilibrium. 

We  have  here  another  case  of  the  simplification  of  mo- 
tion. Instead  of  pursuing  an  irregular  motion  with  two 
independent  precessions,  the  poles  fall  into  step  180° 
apart,  and  the  motion  is  afterwards  performed  as  if  the 
moon  and  her  orbit  were  rigidly  connected,  moving  to- 
gether as  a  whole  in  the  symmetrical  position  where  the 
moon's  axis,  the  axis  of  her  orbit  and  the  axis  of  the 
ecliptic  all  lie  in  one  plane.  The  moon's  axis  thus  moves 
with  a  nearly  constant  precession  and  with  a  nearly  con- 
stant inclination  to  the  pole  of  the  ecliptic  —  practically 
a  Poinsot  motion,  or  a  motion  under  the  action  of  no 
forces. 


EFFECT   ON   HER   OWN   AXIS  95 

The  same  regulatory  couple  exists  for  the  earth's  axis  in 
the  problem  previously  considered,  and  there  is  a  tendency 
to  force  the  earth's  precession  to  keep  step  with  the  pre- 
cession of  the  moon's  orbital  pole,  but  the  masses  being 
reversed,  the  regulatory  couple  is  unable  to  force  the 
earth's  axis  to  the  proper  velocity  and  it  constantly  lags 
behind.  The  distortion  of  the  curve  in  Fig.  24  plainly 
indicates  the  effort  which  the  orbital  pole  makes  to  carry 
the  earth's  axis  along  with  itself. 

This  peculiar  motion  of  the  moon's  axis  has  been 
exactly  confirmed  many  times  by  observation.  Cassini 
first  discovered  it  in  1675  by  observation,  and  it  is  known 
as  Cassini's  Theorem.  It  is  usually  stated  thus:  "The 
plane  of  the  moon's  orbit,  her  equatorial  plane,  and  a 
plane  through  her  centre  parallel  to  the  ecliptic,  always 
intersect  in  the  same  line,  and  the  ecliptic  plane  always 
lies  between  the  other  two." 

In  all  revolving  systems,  both  the  central  and  the  satel- 
litic  bodies  always  tend  to  fall  into  a  Cassini  motion,  and 
the  smaller  bodies  generally  acquire  such  a  condition  at  an 
early  stage.  It  is  quite  certain  that  all  the  nearer  satellites 
of  our  system  execute  Cassini  motions,  and  from  tidal 
forces  all  the  nearer,  and  probably  also  the  remoter  ones, 
perform  their  rotations  and  revolutions  in  the  same 
period.  There  is  however  no  connection  between  the  two 
phenomena  except  in  so  far  as  a  low  rotational  velocity 
favors  the  action  of  the  regulatory  couple. 

A  certain  historical  interest  attaches  to  Cassini's 
theorem.  Shortly  after  this  peculiar  motion  was  discovered, 
it  was  perceived  that  there  must  be  some  cause  and 
an  explanation  was  eagerly  sought.  In  1754,  D'Alembert 
attempted  a  solution  without  success.  Finally  in  1764  the 
French  Academy  offered  a  prize  for  the  discovery  of  the 
cause  and  this  prize  was  won  by  Lagrange  in  1780.  His 
solution,  however,  was  only  a  partial  one.  Lagrange  proved 
that  if  the  moon  is  triaxial  with  the  axis  of  least  moment, 
always  pointing  nearly  towards  the  earth,  then,  such  a 


96  NATURAL   PHILOSOPHY 

condition  of  the  three  planes  once  existing,  it  would 
persist.  Routh,  in  his  "Advanced  Dynamics,"  has  given  a 
proof  along  similar  lines.  Both  of  these  proofs  postulate 
that  the  axis  of  least  moment  shall  always  point  approxi- 
mately towards  the  earth.  But  we  have  seen  that  the 
coincidence  of  the  rotational  and  revolutional  periods  is 
not  essential  and  the  body  need  not  be  triaxial.  In  fact  we 
have  supposed  the  moon  to  be  biaxial. 


36.   Glacial  Epochs 

We  have  seen  that  two  bodies  revolving  about  their 
common  centre  of  inertia  are  subject  to  tidal  forces 
tending  to  tear  each  body  apart  from  its  centre  towards 
and  away  from  the  other  body.  They  are  thus  lengthened 
in  the  direction  of  the  line  between  them  and  compressed  in 
a  direction  perpendicular  to  their  orbits.  If  the  rotation 
and  revolution  are  not  the  same,  the  matter  of  the  body  in 
rotating  through  these  body  tides  is  subjected  to  a  kneading 
process  which  reduces  its  rotational  velocity  by  transform- 
ing rotational  energy  into  heat.  In  the  case  of  the  earth 
these  body  tides  are  not  inappreciable.  They  are  slight 
but  they  certainly  exist,  and  the  continuous  operation  of 
even  a  slight  action  for  immense  periods  of  time  has,  as 
we  shall  see,  far-reaching  effects. 

The  earth  is  rotating  at  all  times  about  two  axes  —  the 
polar  axis  and  the  precessional  axis  which  is  perpendicular 
to  the  former.  As  these  rotations  have  to  be  executed 
through  the  tidal  distortion,  they  are  constantly  being 
opposed.  The  angular  velocity  of  the  tide  is  the  orbital 
angular  velocity,  a,  of  the  attracting  body  —  sun  or 
moon  —  and  for  both  bodies  the  average  axis  of  the  tide  is 
perpendicular  to  the  ecliptic.  If  H  is  the  tidal  effect,  or 
couple,  we  may  suppose  it  decomposed  into  two  tides, 
H  cos  d^  about  the  diurnal  axis  and  H  sin  i?  about  the  pre- 
cessional axis.  The  effect  of  either  tide  is  proportional  to 
the  difference  of  the  tidal  and  rotational  velocities,  or 


GLACIAL   EPOCHS  97 

the  couple  about  the  diurnal  axis  is  proportional  to  H  cos  t? 
(a  —  co),  while  that  about  the  precessional  axis  is  pro- 
portional to  H  sin  ??  (a  —  ^  sin  d).  The  former  couple  is 
negative  and  tends  to  reduce  w  to  a,  while  the  latter  is 
positive  and  tends  to  reduce  the  negative  precession, 
yj/  sin  t?. 

The  constant  negative  precessional  velocity,  yj/  sin  t?, 
which  is  just  sufficient  to  balance  the  average  gravitational 
couple  and  maintain  the  inclination  constant,  we  have 

found  to  be  —  j^  sin  t?  cos  t?.  If  we  reduce  this  negative 

velocity  by  braking,  or  accelerate  positively,  it  will  no 
longer  be  able  to  support  the  gravitational  couple  and  the 
axis  will  yield  in  part  to  this  couple.  In  the  case  of  a  top, 
which  is  precisely  similar,  if  we  reduce  the  precession  by 
the  slightest  amount,  it  begins  to  fall,  and  if  we  abolish 
the  precession  altogether  it  falls  exactly  as  if  there  were  no 
rotation.  The  effect  of  the  tidal  brake  is  that  the  earth 
never  has  quite  the  full  amount  of  precession,  or  the  free 
precession,  necessary  to  maintain  its  inclination  constant, 
and  the  axis  slowly  falls  away. 

We  may  therefore  divide  the  motion  into  two  parts, 
viz.,  the  actual  precession  combined  with  nearly  all,  but 
not  quite  all,  of  the  gravitational  couple,  resulting  in  a 
precessional  motion  with  constant  inclination,  together 
with  an  extremely  minute  gravitational  couple  which  is 
unbalanced  by  any  precession  and  which  results  in  a 
pendulation  through  the  pole  of  the  attracting  orbit, 
exactly  as  if  there  were  no  rotation. 

Regarding  it  from  another  point  of  view,  we  may 
consider  the  actual  precession  to  be  a  free  precession  to- 
gether with  a  minute  precession  in  the  opposite,  or  posi- 
tive, direction,  the  algebraic  sum  of  the  two  being  the 
actual  precession.  The  motion  can  thus  be  divided  into 
two  parts  —  the  free  precession  which  exactly  balances  the 
gravitational  couple  and  which  would  maintain  the  in- 
clination with  no  other  forces,  together  with  a  minute 


98  NATURAL   PHILOSOPHY 

direct,  or  positive,  precession  unbalanced  by  any  gravita- 
tional couple,  which  causes  the  axis  to  pendulate  through 
the  pole  of  the  ecliptic,  exactly  as  in  the  case  of  the  gyro- 
scopic compass.  Whether  regarded  as  an  ordinary  gravita- 
tional pendulum,  or  as  a  gyroscopic  pendulum,  the  motions 
are  equivalent. 

Considering  the  earth  as  absolutely  rigid,  its  axis 
would  precess  in  a  small  circle  about  the  pole  of  the 
ecliptic  at  a  constant  average  inclination  forever.  But  the 
sweep  of  the  tide  is  equivalent  to  a  minute  positive  couple 
about  an  axis  ±  to  the  ecliptic,  with  the  result  that  the 
precessional  curve  is  not  exactly  re-entrant  but  gradually 
spirals  in  towards  the  pole. 

Let  us  consider  the  following  problem  which  is  similar 
to,  but  not  identical  with,  the  actual  case.  We  shall  con- 
sider the  earth  to  be  absolutely  rigid  (non-def ormable) , 
and  a  constant  positive  couple,  H,  is  applied  about  an 
axis  through  its  centre  perpendicular  to  its  orbit.  Let  co 
be  the  rotational  velocity  of  the  earth  at  the  beginning  of 
an  epoch  and  a>3  that  at  any  subsequent  time.  The  average 
gravitational  couple  for  a  complete  revolution  we  have 

found  to  be  —  -r-  sin  ^  cos  t?,  or  half  the  maximum  couple. 

We  can  effect  the  same  average  precessional  motion  by 
placing  the  half  mass  of  the  attracting  body  at  the  pole  of 
its  orbit  at  a  distance  equal  to  its  average  distance,  and 
supposing  it  to  repel  instead  of  attracting. 
The  equations  of  motion  are, 

-  y  sin  t?  cos  t?  -  Cco3  i/'  sin  t?  +  Ayp^  sin  t?  cos  t?  =  A^  (1) 

if  sin  T>  +  Coi3  ^  -  A>P  cos  m  =  ADt  (rp  sin  t>)     (2) 
Hcost?  =  CDtc^s  (3) 

From  (2)  and  (3)  we  derive  the  momental  equation, 

Ht  =  Cco3  cos  i^  -  Ceo  cos  t?o  +  A\l/  sin2  t?  (4), 
which  states  that  the  increase  of  the  moment  of  momentum 
about  the  axis  of  the  couple  is  measured  by  the  time 
integral  of  the  couple.  From  (4), 


/ 


GLACIAL   EPOCHS  99 

ZJf2  C2 

^  Sin2  Mt    =   -^ Jrf    (^3^   -   0)2)    +  Ceo  COS  t?o^, 

C(x)3  COS  t?(i^    =    ^^  (CO32    —  co2) . 

From  (1)  and  (2)  we  have  the  energy  equation, 

■J  (sin2  t?o  -  sm2 1?)  +  -^j-  -  ^  (C032  -  w2)  H j—±  H  = 

y   (t?2  +   (^Sin^)2).  (5). 

It  is  evident  that  the  axis  spirals  in  towards  the  pole  with 
a  negative  precession  and  at  the  pole,  Ht  =  C{ca3  —  w  cos  t?o), 
while  the  value  of  t?2  at  the  pole  is 

t?2  =  ^  sin2  t?o  H-  ^  co2  sin2  t?o. 

Since  K'  is  small  compared  with  w,  we  may  write 

C 
t>  =  -J  CO  sin  t?o. 

Thus  the  polar  value  of  t>  depends  only  upon  co  and  is 
independent  of  any  intermediate  values,  C03,  of  the  rota- 
tional velocity.  If  there  were  a  couple  retarding  the  rota- 
tional velocity,  as  in  the  actual  case,  instead  of  an  ac- 
celerating couple,  the  polar  value  of  t?  would  be  the  same. 
We  shall  now  start  from  the  pole,  as  a  new  epoch,  with  a 
velocity, 

C 

&  =  -T-  (i)  sin  i?o- 

Let  o)p  be  the  rotational  velocity  at  the  beginning  of  this 
epoch.  The  momental  equation  is  now 

Ht  =  Cco3  cos  d^  -  Co)p  +  A\l/  sin2  ??  (6). 

It  is  evident  that  as  the  axis  spirals  outwards  the  preces- 
sion will  be  direct,  or  in  a  positive  direction.  Equa.  (5) 
now  becomes 

y^^2  -^sin2  t?o)  +y (v^sint?)2.  (7). 


100  NATURAL   PHILOSOPHY 

Putting  the  value  of  Ht  from  (6)  in  (7)  and  making  t?  =  0, 

Ayj/  sin  t?  cos  t>  =  Cco3  sin  t?  =±=    OcoZ  sin2  t?o j-  sin2|?  V   (8) . 

Since  K  is  small  compared  with  oj,  we  can  write 

i4^  sin  t?  cos  t?  =  Caj3  sin  t?  =*=  Ceo  sin  t?o.  (9). 
This  is  the  condition  for  the  extreme  outward  swing.  It  is 
evident  that  the  precession  will  still  be  positive  when  4  be- 
comes zero.  4/  sin  t?  cos  t?  and  C03  sin  t?  are  the  components  in 
the  orbital  plane  of  the  rotational  velocities  about  their 
respective  axes.  These  components  are  about  the  same 
axis  but  in  opposite  directions  and  therefore  have  different 
signs.  If  we  consider  1/'  sin  d^  cos  ??  negative,  then  0)3  sin  t?  is 
positive  and  for  the  right  member  of  (9)  to  be  negative  we 
must  use  the  lower  sign  and  Cwz  sin  t?  <  Ceo  sin  t?o-  If  we 
consider  ^j/  sin  t?  cos  d^  positive,  then  Cco3  sin  t?  is  negative 
and  for  the  right  member  to  be  positive,  we  must  use  the 
upper  sign  and  CC03  sin  d^  <  Cm  sin  t?o-  But  C03  >  co,  whence 
sin  ^  <  sin  t?o-  Consequently  the  axis  starts  on  its  second 
swing  towards  the  pole  from  a  nearer  position  than  on  the 
first  swing.  It  will  be  noted  that  in  Equa.  (9)  the  rotational 
velocity  03 p  for  the  beginning  of  the  epoch  does  not  appear. 
Consequently  if  during  the  outward  swing  the  rotational 
velocity  were  retarded,  as  is  actually  the  case,  instead  of 
being  accelerated,  the  motion  would  be  similar.  Any 
variation  of  the  velocity  about  the  polar  axis  does  not 
influence  the  direction  or  general  nature  of  the  motion 
about  the  other  axes:  it  merely  modifies  slightly  the 
amount  of  such  motions. 
Making^  sin  t?  =  0,  we  have 
C 


'-       A 


(co2  sin2  t?o  -  W32  sin2  ^)  V. 


The  point  where  the  precession  becomes  retrograde  is 
therefore  within  the  original  starting  point  and  it  is  evident 
that  the  axis  will  finally  come  to  rest  _L  to  the  orbit. 

The  motion  is  represented  diagrammatically  in  Fig.  25. 
The  full  curve  represents  the  retrograde  spiral  inward 
and  the  dotted  curve  the  direct  spiral  outward.  At  A  the 


GLACIAL   EPOCHS  101 

axis  has  ceased  going  outward  and  at  B  the  precession 
becomes  retrograde.  Each  inward  spiral  is  begun  succes- 
sively nearer  to  the  pole.  If  the  couple,  H,  is  very  small, 
the  period  of  each  swing  is  very  great. 

The  actual  case  of  the  earth,  while  generally  similar, 
differs  in  some  respects  from  the  preceding  problem.  The 
diurnal  rotation,  instead  of  being 
accelerated,  is  retarded,  but  the  ,,.J^ 

precessional    rotation   is    acceler-  ■>^<jmr7*\*»^ 

ated,  as  in  the  problem.   We  do        /^><!^^^^rSv\\'^ 
not  know  the  exact  expressions  for       //'///'Ciir\*\\\\\ 
the  couples  about  the  two  axes,       I  M  •  f:''<^  U  H  i  I  I 
and  if  we  did  the  equations   of       \\  \^^  V^O-^'V// / 
motion  would  probably  not  be  in-       V\  vV>--*^^^v'* 
tegrable.     However,    the    general  >v»o--IIII^--^/ 

nature  of  the  motion  is  evident  ^^^    1 

and  it  must  be  like  that  in  the  pre-  ^^  ^^  Q 

ceding  problem.  In  Equa.  (1)  the  A 

factors  governing  the  polar  motion  pj^^  25 

of  the  axis  are  the  gravitational 

couple  and  the  gyroscopic  couple,  —  Ccoa^  sin  t?.  When  the 
precessional  and  diurnal  motions  are  both  accelerated  it  is 
evident  that  the  axis  gets  nearer  to  the  pole  with  every 
swing.  However,  whether  the  amplitudes  of  the  swings 
successively  decrease  or  increase  depends  upon  the  relative 
variation  of  the  two  factors,  ws  and  i/'  sin  t?,  —  the  rotations 
about  the  two  axes.  If  003  decreases  in  a  relatively  greater 
proportion  than  the  precession  is  accelerated,  the  am- 
plitude of  each  swing  will  become  progressively  greater 
until  the  positive  end  of  the  axis  of  a  planet  may  be  brought 
to  the  other  side  of  the  ecliptic,  and  its  diurnal  rotation 
will  appear  to  be  retrograde. 

We  may  consider  the  nearer  satellites  of  a  planet  as 
rigidly  attached,  dynamically,  to  its  equatorial  plane,  so 
that  the  orbits  of  the  nearer  satellites  will  be  "tipped  over" 
with  the  planet,  and  their  revolutions  will  appear  to  be 
retrograde.  The  fact  that  such  motions  occur  in  our  solar 


102  NATURAL   PHILOSOPHY 

system  is  therefore  not  an  argument  against  the  nebular 
hypothesis.  The  less  the  density  and  rigidity  of  a  planet, 
the  more  likely  is  it  that  its  diurnal  rotation  will  be  slowed 
down  disproportionately  to  the  tidal  acceleration  of  its 
precession,  resulting  in  an  extreme  pendulation.  If,  in  the 
case  of  the  gyroscopic  compass,  the  precession  remains 
practically  constant  while  the  rotational  velocity  steadily 
decreases,  an  extreme  pendulation  will  result,  even  through 
the  opposite  pole. 

The  problem  as  treated  here  does  not  take  account  of  the 
fact  that  besides  the  breaking  action  of  the  tide,  the  earth 
is  distorted^  so  that  the  tidal  mass  is  actually  performing  an 
independent  rotation  in  the  plane  of  the  ecliptic.  This 
gives  rise  to  a  moment  of  momentum  about  the  axis  of  the 
ecliptic  equal  to  the  mass  of  the  tide,  into  the  square  of  its 
distance  from  the  centre,  into  the  orbital  velocity.  This  is 
the  same  as  if  the  earth  were  perfectly  rigid  and  had  an 
added  moment  about  the  orbital  axis  converting  it  into 
a  gyroscopic  compass.  The  effect  is  very  slight,  but 
additive  to  the  breaking  effect. 

If  the  earth  were  a  perfectly  rigid  body,  it  would  main- 
tain its  inclination  constant,  but  perfectly  rigid  bodies  do 
not  exist.  The  motion  of  an  elastic  body  is  different  and, 
given  sufficient  time,  the  results  are  widely  different. 
There  is  not  the  least  doubt  that  the  earth  executes  pendu- 
lations  through  the  pole  of  the  ecliptic,  and  there  is  abun- 
dant evidence  that  it  has  executed  several  such  swings  in 
the  past.  The  extent  of  glaciation  about  the  poles  at  any 
time  is  simply  a  function  of  the  axial  inclination.  With  the 
axis  J_  to  the  ecliptic  there  would  be  no  ice  anywhere  and  a 
genial  climate  would  exist  even  at  the  poles.  There  is 
undoubted  evidence  that  a  subtropical  flora  flourished  near 
the  poles  in  a  comparatively  recent  past.  This  is  positive 
proof  that  the  axis  at  that  time  had  little  inclination, 
for  this  flora  could  not  have  flourished  without  continuous 
light  as  well  as  heat.  With  the  present  inclination  there  are 
extensive  ice  caps  at  the  poles  and  with  a  few  degrees  more 


GLACIAL   EPOCHS  103 

of  inclination  these  caps  would  extend  to  twice  their 
present  area,  as  happened  during  the  last  extreme  glacia- 
tion. 

The  period  of  the  swing  must  be  enormously  great. 
What  it  is  we  do  not  know  and  perhaps  centuries  must 
elapse  before  we  have  any  definite  knowledge. 

It  is  stated  that  Eratosthenes  (B.C.  250)  found  the 
inclination  to  be  23°  51'  20"  and  that  Hipparchus  (B.C. 
120)  found  it  23°  51'.  The  determination  of  this  angle 
requires  a  high  state  of  contemporaneous  civilization. 
Only  a  few  centuries  ago  such  determinations  were  no- 
where possible.  Shortly  before  and  after  the  Christian  era 
there  were  astronomers  who  could  attempt  it.  Going  far- 
ther back  we  come  to  another  stage  of  barbarism  and 
beyond  this  in  a  very  remote  antiquity,  we  come  to  the 
Pyramid  builders,  who  were  astronomers  of  a  high  order. 
Their  time  has  been  estimated  all  the  way  from  3000  to 
13,000  B.C.  and  even  more,  but  the  simple  fact  is  we  do 
not  know  when  they  lived. 

These  builders  have  left  us  a  peculiar  angle  in  their 
oldest  pyramids.  This  angle  is  26°  and  its  double  52°. 
The  slopes  of  all  the  faces  are  52°  and  the  inclination  of  all 
the  passages,  whether  descending  or  ascending,  is  26°. 
The  selection  of  this  particular  angle  and  its  constant  re- 
petition could  not  have  been  accidental.  They  undoubtedly 
had  measured  the  inclination  of  the  earth's  axis  to  within  a 
minute.  This  is  the  one  great  angle  in  all  nature  which 
would  impress  itself  upon  intelligent  men.  There  is  no 
other  predominant  angle  for  an  earth-dweller.  There  is  a 
presumption  then  that  this  angle  was  the  inclination  of  the 
earth's  axis  at  that  time  and  that  its  double,  or  52°,  was 
the  breadth  of  the  then  tropical  zone.  But,  of  course,  this 
is  only  a  surmise.* 

*r.  Popular  Astronomy.  Dec.  1916. 


104  NATURAL   PHILOSOPHY 


37.     The  Earth's  Surface 


The  rigid  demonstration  of  a  mathematical  proposition 
is  something  which  must  be  eternally  true.  There  are, 
however,  many  cases  where  we  cannot  hope  to  arrive  at 
exact  results,  but  only  at  probable  truths,  or  even  at 
possible  truths.  It  is  thus  legitimate  to  speculate,  pro- 
vided we  always  keep  carefully  in  mind  the  distinction 
between  exact  knowledge  and  surmise.  In  the  present 
article  we  shall  allow  ourselves  to  speculate  upon  what 
part  known  forces  may  have  had  in  shaping  the  earth's 
surface. 

The  earth  is  progressively  denser  from  the  surface  to  the 
centre,  and  layers  of  equal  density  are  ellipsoids.  The 
ellipticity,  or  deviation  from  sphericity,  of  these  surfaces 
increases  from  the  centre,  where  it  is  zero,  to  the  surface. 
The  principal  moments  of  inertia  of  these  shells  vary 
therefore,  and  the  quantity,  C  —  A,  upon  which  the 
precession  depends,  increases  to  the  surface.  The  pre- 
cessional  rates  of  the  different  shells  being  different  and 
the  interior  being  plastic,  the  precession  is  not  executed  as 
a  whole,  as  it  would  be  in  a  perfectly  rigid  body,  but  there 
is  a  tendency  for  some  of  the  outer  shells  to  move  over 
each  other.  Since  it  is  not  possible  for  a  spheroidal  shell 
to  turn  about  its  long  axis  over  an  enclosed  shell  very  far, 
such  motion  must  be  very  limited.  There  result,  however, 
readjustments  which  may  be  smooth  and  regular,  or 
occasionally  effected  suddenly.  Thus  earthquakes  arise 
which,  though  actually  very  slight  movements,  seem  to 
observers  like  men  to  be  of  extraordinary  intensity.  It  is 
evident  that  if  the  earth  were  isolated  such  phenomena 
could  not  arise  and  their  occurrence  points  plainly  to 
the  action  of  external  bodies. 

The  effect  of  the  shifting  of  large  masses  over  the  earth's 
surface  is  an  interesting  problem.  The  aqueous  vapor  in  the 
atmosphere  is  a  small  but  appreciable  fraction  of  the 
earth's  mass.  Ordinarily  the  weight  of  the  atmosphere  is 


THE   EARTH'S   SURFACE  105 

distributed  in  a  fairiy  even  manner  over  the  earth's 
surface,  but  if  during  the  beginning  of  a  glacial  epoch  the 
aqueous  content  going  poleward  is  locked  up  there  and  not 
allowed  to  return,  a  disturbance  of  equilibrium  results. 
The  first  result  of  a  concentration  of  matter  towards  the 
poles  is  a  diminution  of  the  moment  of  inertia  about  the 
earth's  axis  with  a  corresponding  increase  in  the  rotational 
velocity,  since  the  moment  of  momentum  must  remain 
constant.  A  secondary  result,  which  is  corrective  of  the 
former,  is  a  bulging  of  the  equatorial  regions,  due  to  the 
increased  rotation  and  the  increased  weight  at  the  poles,  as 
the  earth  strives  to  regain  its  former  potential  surface. 
These  effects  are  extremely  slight,  but  an  inequality  may 
remain  uncorrected  for  some  time  and  a  sudden  readjust- 
ment may  result  in  appreciable  effects.  A  moderately 
rapid  change  of  the  rotational  velocity,  by  a  fraction  of  a 
second,  would  result  in  tangential  stresses  which  would 
throw  up  long  north-south  ridges  (mountain  chains).  A 
moderately  rapid  change  of  the  ellipticity  of  the  earth 
would  result  in  the  throwing  up  of  east- west  ridges,  and  as 
these  two  effects  might  occur  simultaneously,  we  may  have 
diagonal  ridges.  Owing  to  the  plasticity  of  the  earth,  it 
keeps  its  surfaces,  both  interior  and  exterior,  very  nearly  in 
an  equipotential  condition  at  all  times.  Deviations  may 
accumulate  for  a  short  time  and  then  be  corrected  sud- 
denly (catastrophically ) ,  but  the  divergence  is  never  wide. 
The  existence  of  ridges  in  the  cardinal  directions  and  their 
diagonals,  strikingly  exhibited  upon  the  earth's  surface, 
points  to  the  dynamical  (rotational)  causes  which  we  have 
just  considered. 

During  the  times  in  the  past  also,  when  the  axis  was  ± 
to  the  ecliptic,  the  earth  must  have  been  subjected  to 
peculiar  and  violent  stresses.  At  this  point  the  angular 
velocity  about  the  4  axis  is  a  maximum,  and  this  axis 
shifts  suddenly  in  the  plane  of  the  equator  to  a  point  180° 
opposite  and  then  back  again  just  before  and  just  after 
the  pole  is  passed.  Or  in  a  comparatively  short  time  the 


106  NATURAL   PHILOSOPHY 

comparatively  large  1}  velocity  is  reversed.  Such  a  catas- 
trophic commotion  must  result  in  extensive  fracturing  of 
the  earth's  crust  and  the  throwing  up  of  east- west  ridges. 

The  older  ideas  that  the  inequalities  of  the  earth's 
surface  were  due  to  the  adaptation  of  its  crust  to  a  slowly 
contracting  (cooling)  core,  are  found  upon  examination  to 
be  untenable.  The  effect  would  be  too  slight  to  produce 
the  observed  phenomena,  and  if  there  were  such  an  effect 
it  would  be  entirely  different. 

We  may  therefore  provisionally  conclude  that  seismic 
disturbances  at  the  present  time  in  all  probability  have 
their  origin  in  the  earth's  precession,  and  that  the  major 
upheavals  of  the  past  probably  were  caused  by  minute 
though  rapid  changes  in  the  rotational  velocity,  and 
especially  at  a  particular  time  when  the  earth's  axis  was  ± 
to  the  ecliptic.  There  was  probably  a  connection  between 
some  of  these  upheavals  and  former  glacial  periods. 

38.     Sufficiency  of  Natural  Forces 

dv 
Starting    with    the  fundamental    law,  f  =  nt  —7- ,   we 

have  derived  all  the  main  principles  of  natural  philosophy 
and  explained  many  of  the  actions  continually  taking 
place  about  us.  The  mathematical,  or  inductive,  method 
employed  is  merely  a  system  of  close  and  careful  reasoning 
—  the  only  one  by  which  absolutely  true  results  can  be 
secured.  Some  of  the  proofs  have  been  given  in  words 
but  are  none  the  less  mathematical  for  that  reason. 
Symbols  are  merely  a  shorthand  for  recording  various 
steps  in  the  reasoning. 

Natural  philosophy  is  not  limited  to  the  scrutiny  of 
particular  problems,  but  should  supply  us  with  an  insight 
into  all  matters  —  even  the  highest.  It  does  not  follow 
that  all  or  any  of  these  higher  problems  will  necessarily 
ever  be  solved,  but  if  they  ever  are  solved  it  must  be  by 
this  method  of  strict  and  careful  reasoning,  or  "organized 
common  sense."  In  all  our  experience  we  have  never  been 


SUFFICIENCY   OF    NATURAL   FORCES  107 

able  to  recognize  more  than  two  things,  viz.,  matter 
and  motion,  or  simply  moving  matter. 

The  question  arises  whether  there  is  something  more 
which  we  have  hitherto  failed  to  recognize  —  a  tertium 
quid.  We  know  that  life  consists  of  moving  matter  and 
that  when  the  motion  ceases  the  system  becomes  dis- 
associated. Was  there  here  something  more?  If  there  was, 
it  was  not  matter  and  it  was  not  force  or  motion,  since 
motion  exists  only  in  connection  with  matter. 

We  have  solved  completely  only  a  few  of  the  simpler 
phenomena  resulting  from  matter  in  motion.  As  we  ad- 
vance in  this  study  we  notice  the  increasing  complexity  of 
the  phenomena.  As  the  factors  increase  the  results  become 
bewildering  and  our  brains  which  are  composed  of  only  a 
limited  number  of  cells  of  matter  in  motion  are  unable  to 
follow  them.  Reasoning  inductively  from  the  complicated 
phenomena  which  a  few  particles  in  motion  can  produce  to 
what  an  infinity  of  such  particles  under  an  infinitude  of 
forces  should  be  able  to  produce,  we  can  impose  no  limit 
to  the  resulting  phenomena.  We  have  no  right,  therefore, 
from  our  experience,  to  deny  that  there  are  any  phenomena 
which  cannot  be  effected  by  matter  in  motion.  We  have 
explored  only  a  part  of  the  field  and  beyond  are  vast 
regions  which  may  always  be  beyond  our  reach,  but  as  we 
progress  there  is  nowhere  any  evidence  of  a  limit  —  only 
an  unlimited  vista  and  increasing  complexity.  The  mathe- 
matical theory  of  probability  projects  a  certain  trend  into 
the  unknown,  often  with  surprising  accuracy,  and  both 
our  experience  and  probability  point  strongly  to  the 
sufficiency  of  matter  in  motion. 

The  tendency  of  mankind,  on  seeing  phenomena  which 
it  does  not  understand,  is  to  ascribe  them  to  supernatural 
agencies,  and  this  attitude  has  an  important  bearing  on  our 
present  inquiry.  Primitive  men  see  in  all  the  phenomena  of 
nature,  spirits,  good  and  bad.  In  the  past,  as  in  the  pre- 
sent, there  have  always  existed  in  the  minds  of  men  hosts  of 
elves,  goblins,  ghosts,  daemons  and  what-not,  who  are 


108  NATURAL   PHILOSOPHY 

continually  performing  supernatural  acts.  These  spirits 
and  their  acts  have  formed  the  bases  of  their  religions  and 
to  deny  that  our  present  religions  are  evolved  from  them  is 
simply  to  deny  that  there  has  been  mental  as  well  as 
physical  evolution. 

When  a  Samoan  is  photographed  he  believes  that  a 
daemon  is  in  the  camera  and  when  a  Hottentot  hears  a 
phonograph  he  has  no  doubt  but  that  a  spirit  is  producing 
the  sounds.  We  have  a  smug  way  of  imagining  ourselves 
very  much  superior  to  populations  of  the  past,  but  the 
difference  is  only  in  degree  and  very  slight  at  that.  We 
have  very  recently  learned  how  to  utilize  a  few  of  the 
forces  of  nature,  which  the  average  man  does  not  under- 
stand, but  on  the  whole  it  is  very  likely  that  the  average 
ancient  Egyptian  was  about  as  intelligent  as  the  average 
man  of  today.  If  an  advanced  intellect  at  some  time  in  the 
future  shall  look  back  upon  both  of  us,  he  will  probably 
find  little  difference.  For  the  most  civilized  and  educated 
among  us  firmly  believe  in  miracles  in  the  past,  if  not  in  the 
present,  and  many  of  our  ideas,  if  analyzed  by  such  an 
advanced  intellect,  would  appear  most  extraordinary. 

The  average  civilized  man  of  today  is  in  advance  of  the 
Samoan  in  that  he  does  not  see  the  necessity  of  having  a 
daemon  in  a  camera,  but  when  it  comes  to  a  monad  they  are 
on  all  fours  and  both  insist  upon  the  daemon.  The  natural 
philosopher,  with  a  fuller  understanding  of  matter  and 
motion  cannot  share  this  belief.  And  he  has  not  the  least 
desire  that  others  should  share  his  view  unless  they  can 
recognize  that  the  probability,  amounting  almost  to  a 
certainty,  is  that  the  camera  and  the  monad  are  both 
phenomena  of  matter  in  motion.  It  is  not  impossible  that 
we  shall  sometime  be  able  to  produce  life  artificially. 

The  vexed  question  of  the  immortality  of  the  soul  is  a 
simple  one  for  the  natural  philosopher.  Under  all  argu- 
ments for  such  an  immortality  there  stands  out  plainly  the 
personal  desire  of  the  pleader  that,  having  existed  for 
some  few  years  as  a  congeries  of  certain  moving  carbon, 


SUFFICIENCY   OF    NATURAL   FORCES  109 

nitrogen,  oxygen  and  hydrogen  atoms,  he  may  somehow 
and  in  some  way  continue  a  very  different  kind  of  existence 
for  all  eternity.  While  all  the  rest  of  nature  is  continually 
undergoing  flux  and  evolution,  he  alone  remains  fixed 
forever!  There  have  been  many  who  have  had  no  desire 
for  such  an  eternal  existence,  but  they  have  based  no 
arguments  upon  their  personal  wishes.  Both  the  matter 
and  the  motion  of  a  living  organism  are  immortal,  but  they 
no  longer  form  the  same  system  after  dissolution.  The 
natural  philosopher  cannot  hold  the  view  that  the  billions 
of  the  earth's  past  populations  —  without  considering  the 
lower  animals  —  still  exist  as  separate  entities,  or  dis- 
embodied souls. 

It  may  seem  that  such  questions  are  wholly  foreign  to 
our  subject,  but  the  domain  of  the  natural  philosopher  is 
the  whole  universe,  and  there  is  nothing  in  it  he  may  not 
philosophize  about,  provided  he  preserves  the  strict 
methods  of  his  science.  To  many,  such  questions  may  seem 
to  be  axioms,  unworthy  of  serious  discussion.  But  it  must 
be  remembered  that  the  great  majority  of  living  men 
firmly  believe  in  miracles  and  are  convinced  that  the 
laws  of  nature  are  not  really  laws,  or  at  best  are  only  laws 
for  a  part  of  the  time,  since  occasionally  they  are  broken. 
It  is  possible  that  in  some  higher  stage  of  advancement 
mankind  may  finally  use  the  means  at  his  disposal  for 
obtaining  the  truth  and  cleave  to  it. 


NOTE 
On  the  Cause  of  Gravitation 

The  ether  is  the  seat  of  an  enormous  store  of  energy  as 
evidenced  by  its  enormous  pressure.  In  the  last  analysis 
this  energy  must  be  kinetic,  or  due  to  some  kind  of  motion 
within  the  ether,  although  we  have  as  yet  not  the  slightest 
conception  of  the  nature  of  such  a  motion.  The  atoms  of 
gross  matter,  being  imbedded  in  the  ether,  necessarily 
partake  of  this  motion  just  as  specks  within  a  liquid  par- 
take of  the  motions  of  the  surrounding  molecules,  con- 
stituting the  well  known  Brownian  movements.  They  are 
thus  foci  which  reflect  and  radiate  the  internal  ethereal 
vibrations. 

The  atoms  being  in  incessant  motion  and  the  ether 
possessing  both  elasticity  and  inertia,  among  other  dis- 
turbances, longitudinal  waves  necessarily  result. 

We  shall  prove  that  such  longitudinal  waves  necessarily 
cause  an  attractional  action  between  all  atoms  of  gross 
matter.  A  longitudinal  wave  is  composed  of  two  halves 
having  opposite  properties.  In  one  half  the  medium  is 
above  its  normal  density  and  its  particles  are  moving  with 
or  against  the  wave  direction,  while  in  the  other  half  the 
medium  is  below  its  normal  density  and  the  particles 
are  moving  in  a  reversed  direction.  An  atom  swept  by  the 
wave  will  therefore  be  urged  alternately  towards  and  away 
from  the  radiating  point. 

Let  V  be  the  average  velocity  of  the  stream  in  either 
direction,  a  the  amplitude  of  the  wave,  or  the  maximum 
distance  any  particle  of  the  medium  moves  from  its 
position  of  equilibrium,  V  the  wave  velocity,  D  the  density 
of  the  medium,  P  its  pressure,  I  a  wave  length,  and  t  the 
time  of  a  complete  vibration.  The  force  with  which  such  a 

110 


ON   THE   CAUSE   OF   GRAVITATION    .  Ill 

stream  urges  an  atom  is  proportional  to  its  velocity,  to  its 
density  and  to  the  surface  which  the  atom  opposes  to  the 
stream.  Taking  the  cross  section  of  the  atom  as  unity, 
its  mass  as  M,  and  the  mass  of  an  equal  volume  of  the 
medium  as  w,  it  is  evident  that  such  a  stream  striking  M 
and  m  at  rest,  will  move  them,  in  the  time  of  half  a  wave, 
distances  inversely  as  their  masses,  or  while  it  moves  m  a 
distance  2a,  just  as  it  moves  any  other  portion  of  the 

medium,  it  moves  M  only  — rj- .  Or  Ms  =  2am,  where  5  is 

the  distance  M  is  moved  during  a  half  wave.  At  the  end  of 
a  half  wave,  therefore,  M  is  2  a  -^^ — r^^ —  behind  m. 

The  time  taken  by  each  half  wave  to  clear  m  is  ^  ,  but 
the  time  taken  by  the  compressed  half  to  clear  M  is 

2V         "^y    MV   / 
while  the  time  taken  by  the  expanded  half  to  clear  M  is 


2V+  ^"[-M^^) 


The  average  stream  pressure  in  either  direction  is  the 
same  and  equal  to  kDv,  k  depending  upon  the  units  chosen. 
But  the  time  during  which  this  pressure  acts  is  unequal 
in  the  two  halves.  There  is  a  net  pressure  acting  for  a 

time  4  a  I  ^  j,  in  the  direction  of  motion  of  the  ex- 
panded half,  during  the  passage  of  every  complete  wave. 
This  is  equivalent  to  a  force  acting  continuously  equal  to 

kDv       .      /M  —  m\       T        ,,  .^  ^.       ,  ^, 

— —  .    4  a  I         yr     I  .    In    all    gravitational   waves   the 

expanded    half    moves    towards    the    radiating    source. 
I 


v        4  a 
Since  jz  =  -y-  ,  the  force  urging  the  atom  contrary  to  the 


wave  direction  is  kD  — j—  .  — r-r- —  .  The  potential  and 


112  NATURAL   PHILOSOPHY 

kinetic  energies  in  a  complete  wave  are  equal,  and  it  can 
be  shown  (v.  "Mechanics  of  Electricity")  that  the  total 
energy  in  a  wave,  per  unit  cross  section,  is 
16  a2  DV2   _  16  a2  P 
I  ~        I       ' 

Calling  this  total  wave  energy,  E,  the  attractive  force 

^  ~   V2'       M       '   t' 

Now  —  is  the  energy  crossing  unit  surface  in  unit  time, 

or  the  flux  of  energy  per  unit  surface.   We  may  call  it 

the  density  of  the  energy  flux.  The  term  "Flux  of  force," 

frequently  used,  is  meaningless.    There  is  no  such  thing 

as  a  flux  of  force,  but  a  flux  of  energy  constitutes  a  force. 

If  we  take  an  equal  volume  of  the  medium  in  place  of  M, 

or  make  M  =  w,  there  is  no  force  and  the  atom  merely 

vibrates  about  its  position  of  equilibrium.  If  the  particle  is 

less  dense  than  the  medium,  the  force  becomes  negative 

and  there  is  a  repellant  instead  of  an  attractive  action. 

It  is  evident  that  a  body  less  dense  than  the  ether  will  be 

driven  up  very  quickly  to  the  limiting  velocity,  V,  when, 

since  it  travels  with  the  wave,  all  further  action  ceases. 

If  M  is  very  much  denser  than  the  ether,  as  is  the  case  for 

k      E 
all  gross  matter,/  =  77-  .  —  ,  or  for  all  gross  matter  oppos- 

ing  unit  surface  to  the  wave,  the  attraction  is  simply  pro- 
portional to  the  density  of  the  energy  flux. 

Considering  positive  and  negative  charges  of  electricity 
as  differentiated  portions  of  the  ether  which  are  respec- 
tively denser  and  less  dense  than  the  normal  ether,  elec- 
trostatic attractions  and  repulsions  necessarily  result  from 
the  causes  just  discussed  {v.  "Mechanics  of  Electricity"). 
At  a  time  when  longitudinal  waves  in  the  ether  were 
denied,  as  in  fact  they  are  today,  Lord  Kelvin  wrote, 
"I  affirm,  not  as  a  matter  of  religious  faith,  but  as  a  matter 
of  strong  scientific  probability,  that  such  waves  (com- 
pressional)  exist,  and  that  the  velocity  of  this  unknown 


ON   THE   CAUSE   OF   GRAVITATION  113 

condensational  wave  is  the  velocity  of  the  propagation  of 
electrostatic  force."  Lord  Kelvin,  Baltimore  Lectures. 

There  is  no  doubt  that  all  atoms  ceaselessly  radiate 
longitudinal  waves.  Such  being  the  case,  universal  at- 
tractions and  repulsions  necessarily  follow.  Hence,  if  any- 
one should  seek  to  explain  gravitation  through  some  other 
agency,  he  would  still  have  this  agency  unavoidably 
coupled  with  it.  Nature  works  in  the  simplest  way  possible. 
She  does  not  employ  multiple  agencies  to  produce  a  simple 
effect.  Further  there  is  no  other  conceivable  agency  by 
which  such  an  action  could  be  effected. 

This  action  of  longitudinal  waves  is  not  confined  to  the 
ether  but  is  a  property  of  longitudinal  waves  in  any 
medium.  It  is  readily  verified  experimentally  in  air 
(sound)  waves.  Gravitational  force  is  the  push,  not  pull, 
of  the  ether  streams  against  the  atomic  surfaces,  and  hence 
is  proportional  to  the  cross  section  opposed.  The  work  of 
some  experimenters  seems  to  show  that  certain  atoms, 
or  at  least  certain  arrangements  of  atoms  (molecules)  may 
have  different  cross  sections  in  different  directions.  Thus 
Heydweiler  claims  that  the  weight  of  a  crystal  of  CuSO^, 
where  the  atoms  are  presumably  oriented,  is  not  the  same 
as  that  of  the  same  mass  in  solution,  where  the  atoms  are 
supposed  to  be  unoriented.  Wallace  claims  that  the 
weight  of  a  given  mass  of  water  changes  after  it  is  frozen, 
i.e.,  crystallized  or  oriented,  but  as  yet  we  have  very  little 
knowledge  concerning  such  matters.  The  weight  of  a  given 
mass  should  vary  with  the  orientation  of  its  atoms  to  the 
field,  if  the  cross-sections  of  the  atoms  vary  with  the 
direction. 


END 


*j^ii'»-!<*;^i'iii?; 


THIS  BOOK  IS  DUB  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

W.UU  BE  ASSESSED   ^°ll';'^'i''^j° ^^^^"^ 
OVERDUE. 


^Ejrrs 


-TWO 


6^p^ 


i«5^^ 


lVlIir^3l959 


LD21-100m-7,'39(402s) 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


